There is dispute among the commentators as to whether or not Ramban meant that seriously or just said it for the debate. R. Yaakov Kamanetzky z"l writes (Emet L'yaakov Genesis 44:18) that Ramban just said it for the sake of the debate. The Chattam Sofer (Orach Chaim 1:16) understands that Ramban was expressing a serious belief, but limited his view to extra-talmudic midrashim.
וכבר כתב רמב"ן בספרו מלחמות חובה כל מה שלא נמצא בש"ס בבלי וירושלמי ומדרשיהם אין אדם מחוייב להאמינו והמאמין יאמין
R. Shaul Lieberman z"l cites
a variety of sources in support of the view that Ramban should be
construed as having meant exactly what he said. He concludes with near
certainty, as he puts it, 'that also in this case, the holy mouth of our master
did not speak a lie, even if it would have been a permissible one.' (Sheki'in (Jerusalem, 1939), 81-3.) (quoted in this article).
The Abarbanel too, understands that Ramban as having expressed his true view, and attaches no qualifications to it. (as noted by @jake)
It should be noted that Ramban does frequently dispute midrashic interpretations in his commentary to the Torah.
Additionally he writes (chiddushim to Yevamot 61b) that we do not respond to words of aggada in the context of rejecting a teaching of the Seder Olam.
ואין משיבין על דברי אגדה דלית לן מאי דאמרינן באגדה ובסדר עולם שרבקה בת שלש שנים היתה באותה שעה
This same expression (ein meishivin al divrei aggada) is used by his student Rashba (Chiddushim to Megillah 15a) in the context of the Tosafists' question on an aggadah in the Gemara that states that Ester was married to Mordechai and became forbidden to him. The Tosafists ask why he didn't divorce her, and answer that the necessary witnesses for the divorce would have caused (or risked) publicizing his relationship with her. Rashba asks that he could have written the get himself thus obviating the need for witnesses. He answers that we don't respond to matters of aggada. This seems to mean that the Midrash may be wrong. (See also his responsa 1:50 and 3:340).
The Meiri uses this expression as well, and definitely means that the Midrash may be wrong. He discusses the opinion in the Gemara (Shabbat 55a) that suffering is always as a result of sin.
אמר רב אמי אין מיתה בלא חטא ואין יסורין בלא עון
The Gemara (55b) concludes that this opinion is mistaken.
ותיובתא דרב אמי תיובתא
The Meiri writes there that it is fundamental to Judaism that everything that happens to man is the result of divine oversight. He continues by writing that even though the Gemara concludes by rejecting this view, the fundaments of Judaism aren't determined by simplistic interpretations of Scripture, or by aggadot, as you know that we do not respond to aggadot.
ואע"פ שנשאר דעת האומר כן בתיובתא אין עקרי האמונות תלויות בראיות של פשוטי מקראות ואגדות וכבר ידעת שאין משיבין באגדה
See also the Machzor Vitry (428) who uses this expression regarding a Midrash about the age of Bil'am at the time he was killed, who notes that "we do not respond to aggadot, and perhaps there is Amoraic dispute in the matter. And Chizkuni (Exodus 4:26) who uses this expression regarding a disagreement between the Gemara and "rov haolam"; i.e. most commentators over whether Moses failed to circumcise his older son, or his younger son. See also the almost identical formulation in Chidushei HaRitva (Avodah Zara 8a), Paneach Raza (parshat B'reshit), Halakhot Ketanot of Mordechai (Menachot ch. HaT'khelt: 940), responsa of Rema (100), responsa of R. Menachem Azzaryah of Fanu (36), and Abarbanel (I Samuel ch. 4) who cites this in the name of "Chazal".
It is thus clear that Maharal erred when he wrote (B'er Hagolah: B'er HaShishi; ch. 15) that the expression (he quotes it from R. Nissim Gaon ) means that we don't question aggadot since they may have hidden meanings:
וזהו בעצמו מה שאמרו אין משיבין בדברי אגדה כי אפשר האומר אותה אמרו על דרך נעלם, ולפיכך אין משיבין ואין מקשין באגדה לפשוטה
As this certainly isn't the intent of the Meiri, and it is almost certainly not the intent of Rashba, or the Machzor Vitry. Lastly it is difficult to reconcile with the usage of the Chizkuni who deals with a Midrash of simple textual analysis. It is difficult to claim that he means that there may be deep meaning to the Midrash about which son Moshe circumcised.
The Mevo Hatalmud (generally printed at the end of tractate berakhot) attributed to R. Shmuel Hanagid (11th century) writes very similarly to the Meiri et al. "one should only learn from it that which seems reasonable...everything else disregard" (translation my own. I rendered "ein somchin aleihem" as disregard).]
Similarly, R. Yehudah Ben Barzilai of Barcelona (11th century) writes in his commentary to the Sefer Yetzira that there are drashot that we disregard "ein somchin aleihem":
לכך דרשו רבותינו בקצת דרשות כי המלאכים עתידים לשאול לצדיקים מה פעל אל...ואע״פ שיש מי שמפרש פסוק זה בנביאים והכי מתראה עיקר פירושו מכל מקום הרבה פנים בתורה ואין מספר להגדיות. ואע״פ שאין מי שסומך על הגדות הללו...וכו
Although in context his exact intent isn't fully clear (to me).
R. Sherira Gaon quoted by the Sefer HaEshkol (Hilchot Sefer Torah) writes:
ואמר מר [רב] שרירא הני מילי דנפקי מפסוקי ומקרי מדרש ואגדה אומדנא נינהו, ויש מהן שהוא כך...והרבה יש שאינו כן, כגון מה שאמר ר' עקיבא דמקושש היינו צלפחד...והם הזכירו דעתו של כל אחד ואחד, ואנו לפי שכלו יהולל איש. וכן...תנחומא...וזולתם, רובם אינו כן, ולכך אין אנו סומכין על דברי אגדה. והנכון מהם מה שמתחזק מן השכל ומן המקרא מדבריהם,
These words that are derived from verses and are called midrashim or aggada are estimations (umdena)...Some are indeed correct, but many are not correct...Therefore we do not rely on aggada...Accept as reliable only those that follow from logic or from the verses. (Also cited in the introduction to Menorat Ha-Maor p. 47)].
R. Hai Gaon cited there writes:
הגדה ומדרש אף על פי שכתובין בתלמוד אם לא יכוונו ואם ישתבשו אין לסמוך עליהם, כי כללינו הוא אין סומכין על ההגדה, אלא מה שקבוע בתלמוד שאנו מוצאין להסיר שבושן ולחזקן יש עלינו לעשות, כי לולא שיש בו מדרש לא נקבע בתלמוד. ומה שאין אנו מוצאין דרך לסלק שבושו נעשה כדברים שאין הלכה. ומה שלא נקבע בתלמוד אין אנו צריכין לכל כך, אלא מעיינין בו אם נכון הוא ויפה, דורשין אותו ומלמדין אותו, ואם לאו אין אנו משגיחין עליו.
Even if Aggadot and Midrashim are written in the Talmud, if they do not make sense disregard them, for we have a principle that we do not rely on the aggadot.
Similarly, Rav Hai Gaon is quoted (Otsar HaGeonim Berakhot; Peirushim: 67) as repeating the assertion that aggadot are not based on tradition, and therefore we do not rely on them. (Note that R. Hai precedes R. Nissim, the source for Maharal's quote about Aggadot. Unlike Maharal who claimed that the reason for not relying on aggadot is that they are cryptic, R. Hai Gaon writes that the reason is that they are not authoritative, and are mere conjecture:
הוו יודעים כי דברי אגדה לאו שמועה הם, אלא כל אחד דורש מה שעלה על לבו כגון אפשר, ויש לומר, לא דבר חתוך
Similarly, R. Abraham the son of Maimonides writes in his Torah commentary (Exodus 14:11) that a midrashic identification of characters in an incident is an estimation, rather than a definite fact.
וזו השערה על ההצעה האומרת כל שאתה יכול לתלות ברשעים תלה
Similarly, the Shiltey Gibborim to Avodah Zarah (6a) writes that some Midrashim state things that they themselves know are untrue as a poetic sort of non-literal interpretation of the text:
הנה תראה מה שדרש ...יעקב אבינו לא מת...והשיב לו מקרא אני דורש, כלומר גם אני יודע שמת אלא אני מתכוין לדרוש את המקרא בכל ענין...ועוד אמרו בתלמוד א"י בפ"ס דנזיר וכי המדרשות אמנה הם דרוש וקבל שכר הא לך הדבר מבואר שלא אמרו חכמים המדרשים על דרך אמונה ועיקר אלא להרבות טעמא למקרא ולדרשו בכל פנים
Rambam (cited and discussed here) writes in his letter on astrology:
I know that you may search and find sayings of some individual sages in the Talmud and Midrashim whose words appear to maintain that at the moment of a man's birth, the stars will cause such and such to happen to him. Do not regard this as a difficulty, for it is not fitting for a man to abandon the prevailing law and raise once again the counterarguments and replies (that preceded its enactment). Similarly it is not proper to abandon matters of reason that have already been verified by proofs, shake loose of them, and depend on the words of a single one of the sages from whom possibly the matter was hidden. Or there may be an allusion in those words...
Thus, he maintains the possibility that the sage was mistaken regarding this non-halakhic matter, but his language leaves open the possibility that he would only suggest this regarding the view of individual sages, but not the consensus of the sages. It is noteworthy, however, that this possibility of the erring sage being an individual view is not backed up with any proof-texts of disputants. This suggests that practically Rambam would be willing to form opinions independent of Chazal even when their statements appear to disagree. It would appear, then, that the appeal to theoretical disputants is a mere formality.
In a similar vein, he writes in a responsum (Blau: II: 458) responding to a reference to a Midrash, that it is a mere aggada, and we do not concern ourselves with aggadot, for are they matters of tradition, or are they reasonable? Rather, they are the personal musings of individuals that do not relate to law and are therefore not to be reckoned with. This applies whether or not the Midrashim are recorded in the Talmud.
ולעניין יוצאי תיבה כל אותן הדברים דברי הגדה ואין מקשין בהגדה וכי דברי קבלה הן או מילי דסברא אלא כל אחד ואחד מעיין בפ[ירושן?] כפי מה שיראה לו בו ואין בזה לא דברי קבלה ולא אסור ולא מותר ולא דין מן הדינין ולפיכך אין מקשים [בהן]. ושמא תאמר לי כמו שיאמרו רבים וכי דברים שבתלמוד אתה קורא הגדה כן כל אלו הדברים וכיוצא בהן הגדה הן בענינם בין כתובין בתלמוד בין שהיו כתובין בספרי דרשות בין שהיו כתובין בספרי אגדה.
Radak (Joshua 5:14) goes so far as to state that a Midrash is based on faulty recollection of the verses:
ועוד כי כותב הדרש הזה טעה בפסוק וילן, כי שני פסוקים הם
He is quoted without objection by Rabbenu Avraham ben Sh'lomo in his commentary to Joshua (p. 28).
Similarly, R. Eliyahu Del Medigo (15th century) writes in Bechinat Hadat (Viena, 1833 pp. 55-56):
התלמוד נחלק לב׳ הלקים, הלק בהודעת הדינים כלם וחלק במדרשות ואגדות. והחלק הראשון אין ספק אצל כל בעל דת מאנשי אומתנו שאין ראוי לחלוק עליו כלל כאשר כבר נאמר, אולם החלק השני הוא אשר יתכן לפעמים שלא נסכים בו ולא יקרה בזה חטא, וזה כי התורה לא חייבתנו לשמוע לחכמים אלא בעניני הדינים אשר בם מעשה או בענייני שרשי האמונדה המוסכמים מהם
That is: the Talmud is divided into two parts: the part that includes all the laws, and the part that includes Midrashim and Aggadot. Regarding the former, there is no doubt among every practicing member of our religion, that it is improper to disagree with it whatsoever. However, the second part is the part which it is conceivable to sometimes disagree with, and this is not a sin. This is because the Torah only required us to listen to the Sages regarding practical laws and the universally agreed upon fundaments of faith.
Similarly, R. Samson Rahael Hirsch wrote that aggadic statements are not part of Torah Shebaal Peh from Sinai, and that we should not part from the great authorities such as Rav Sherira Gaon, Rav Hai Gaon, Rabbeinu Nissin, Rabbeinu Chananel, Rabbeinu Shmuel HaNaggid, Ritva... who taught that we do not accept aggada unconditionally, but only accept that which seems reasonable. Furthermore, claiming the opposite, actually opens the doors to heresy!
Similarly, Hakham Yosef Faur writes in Tradition (vol. 9 no. 4):
Traditionally for normative Judaism only the halakhic elements of the Talmud are authoritative.
There is possibly Talmudic precedent for this general approach to aggadot from the Palestinian Talmud Sanhedrin (10:1) as cited by Rashbats (Magen Avot to Tractate Avot 2:14):
ואמרו בירושלמי שם, אבל ספרי המירם והספרים שנכתבו מכאן ואילך, כל הקורא בהם כקורא באגדות, מאי טעמא, 'ויותר מהמה בני הזהר עשות ספרים הרבה אין קץ ולהג הרבה יגיעת בשר' [קהלת יב יב], להגיון נתנם ולא ליגיעה נתנם
Regarding books that are written from know on, reading them is like reading aggadot. Why? [because it is written "of making many books there is no no end," and] "much study is weariness of the flesh" (Eccless 12:12), for reading they were given but for toil [i.e. study], they were not given! (Trans. R. Dr. Jose Faur Golden Doves With Silver Dots p. 100)
Given the overwhelming body of literature expressing the view, including the writings of Ramban's predecessors, contemporaries, successors, and his own writings, it seems that there is no evidence that he believed anything other than what he wrote, and it is reasonable to assume that the Abarbanel and R. Lieberman are correct in their understanding.
Notably, Ramban's argument about the non-authoritative role of Midrashim was repeated in the Disputation at Tortosa by R. Astruc Halevi.
Others, however take the stance that one must believe (in at least some) Midrashim. This is the opinion of R. Soloveitchik according to R. Moshe Meiselman in his Torah Chazal & Science (see p. 654) regarding the source for the dirt used to create Adam, the story of Avraham being thrown into the fiery furnace, and Adam's sacrifice the day he was created. R. Hershel Shechter attributed similar sentiments to R. Soloveitchik, in a conversation I had with him.
The Chazon Ish (Iggerot Chazon Ish I:15) also regards as dogma the belief that all aggadot in the Talmud have their origin in the sages' prophetic power.
משרשי האמונה שכל הנאמר בגמ'...בין בהלכה ובין באגדה, הם הם הדברים שנתגלו לנו ע"י כח נבואי
Somewhat of an intermediate approach seems to be adopted by R. Yechiel of Paris in his Disputation in Paris in which he writes the following in response to the question of whether he believes in the Talmud from which questions will be raised:
Included there are fantastical things which are difficult for heretics to believe, and regarding them there is no need to respond to you. If you wish, believe them, and if you don't want to, don't believe them. For law isn't determined on their account. However, I know that the scholars of the Talmud didn't write mistakes, but only correct and true things. And if they seem fantastic to listeners, [bear in mind]: are there not many similar things in Scripture, such as Lot's wife who became a pillar of salt, and the mouth of Bilam's ass which spoke with him...and many similar events.
ויש בהם דברי פלא שקשה להאמין לכופר ולאפיקורוס ולמין, ועליהם אין צורך להשיבך, אם תרצה תאמין ואם לא תרצה לא תאמין בהם. כי אין משפט נכרת על פיהם. אמנם ידעתי כי החכמים בתלמוד לא כתבו מאומה רק דברים כנים ואמתים, ואם מופלאים הם לשומעיהם הלא יש כאלה רבים במקרא, כמו אשתו של לוט שהייתה נציב מלח, ופי האתון של בלעם שדיבר עמו ומלחמת סיסרא שנלחמו הכוכבים ממסילותם. ומעשה יונה וקיקיון של יונה שבין לילה היה ובין לילה אבד, ומעשה דאלישע שהחיה את המת מקברו וכיוצא בהם רבים
That is although he personally believes that all the words of the Talmud, even the Midrashim, are true and wise, and he implies that the only ones who have difficulty accepting them are heretics, he nevertheless writes that they do not not need to be accepted as they do not determine law.
(As with Ramban's dispute there is some latitude to suggest that some or all of what he said was a result of the necessity of the debate and doesn't reflect his true view).
See also here and Rabbi Chaim Eisen's article in Hakirah vol. 4 at length. Significantly, R. Eisen writes that he is unaware of any Geon or Rishon who held of Maharal's view.