Assume for the purposes of the question that all circumstances are the same as if the wine had been made/handled by non-Jews. I don't want answers regarding whether it is Mevushal or whether Mevushal really helps or if it was opened or not opened, etc. I am asking purely: Are the rules the same for non-religious Jews as they are for non-Jews with regard to making/handling wine? If I cannot drink a bottle of wine because of the way/fact that it was made/handled by a non-Jew, am I prohibited from drinking that bottle if it was made/handled the same way by a non-religious Jew?
To add to Shmuel Brin's answer:
R' Moshe Feinstein rules that a Jew married to a non-Jew is presumed to not observe Shabbos, and R' Moshe implies that such a person would have the presumptive status of a public Shabbos violator. Accordingly, R' Moshe rules that non-mevushal wine such a person touches is forbidden (Igros Moshe YD vol. 2, §132).
R' Yosef Messas (Otzar HaMichtavim 2:1302, h/t @Maimonist) discusses a circumstance in which he rules that the wine of public Shabbos violators would be permitted:
Although R' Messas is generally lenient when it comes to the laws governing wine of non-Jews (such as to permit liquor made from non-Jewish wine, see Otzar HaMichtavim 1:454,462), he affirms that public Shabbos violators would generally render wine forbidden through contact.
However, after noting the abundance of Shabbos violators and the great degree to which the communal life of religious and partially religious (but Shabbos violating) Jews was intertwined,1 R' Messas makes an allowance for the wine. The reason he gives is that prohibiting the wine would be taken as a grave insult that would disrupt the great communal unity in their communities and instead create massive strife. He adds that prohibiting their wine would cause the Shabbos violating Jews to be resentful and totally abandon Judaism and any religious observance. He also mentions that provoking the animosity of the less observant Jews could lead to a severely damaging backlash against the Shabbos observant Jews.
R' Messas' would apparently extend this allowance to any community where treating the wine of Shabbos violators as prohibited would cause this sort of discord, as he mentions that one may actively countermand a rabbinic prohibition for the sake of avoiding strife.2
2 One may actively countermand a rabbinic prohibition for the sake of kavod hab'riyos ("human dignity"; see, for example, B'rachos 19b). R' Messas rules that one may do the same in order to avoid strife, but the basis for this is less clear. R' Yehuda Segal (Tzemach Yehuda 3:59), for example, appears to rule that this allowance does not extend to actively overriding a rabbinic law to avoid strife: "וכבוד הבריות שנדחה היינו באם בזה גופא יש כבוד הבריות ולא שהוא יעשה מחלוקת ואיהו עיוות נפשיה". The Ritva (Shabbos 23b) says that peace and harmony alone allow one to use candles for lighting the home instead of lighting Chanuka candles (thus failing to perform a rabbinic obligation by default), but even the Ritva doesn't go so far as to say that one may actively violate a rabbinic law for peace.
On the other hand, the Sho'eil U'Meishiv (vol. 2, 3:106, "וגדול כבוד הבריות ובפרט בזמנינו קשה המחלוקת") might possibly be understood to halachically link the two. Likewise, it is possible that the Maharam miRothenburg may be understood to permit violating most rabbinic restrictions for the sake of peace (§758, where he upholds the rabbinic prohibition against annulling oaths made in HaShem's name even in the face of an impending divorce. It seems that his reasoning is that there was a particular exigency in upholding that rabbinic restriction - since people were being too flippant with oaths - that outweighed the special importance of fostering domestic harmony).
In order to avoid situations of mixed-marriages , which almost certainly lead to assimilation , the Rabbis enacted many decrees whose goal was to inhibit the forming of close relationships between Jews and non-Jews. The laws of stam yeinam (wine that was produced or handled by Non-Jews) are included in these decrees. On the other hand, the Torah, and in its footsteps, the Rabbis, wanted to encourage close relationships amongst Jews, and they therefore established that a single witness is believed regarding prohibitions. The Yerushalmi explains that this was done in order to enable Jews to conveniently eat and drink with their fellow Jews. Let us therefore preface our response with some of the principles and basic laws regarding wine and when a non-Jew comes in contact with it, only afterwards will we deal with the issue of wine that comes in contact with a Jew. Even if a non-Jew opens a bottle of wine, the contents are not prohibited . If a non-Jew pours a cup of wine, according to many opinions, the wine is prohibited , while some opinions are lenient if a monetary loss, even a small one, is involved . There is also a dispute regarding the permissibility of the wine that remains in the bottle . Even those who are stringent will permit the wine if a large loss would otherwise be incurred . All this is regarding natural wine, but cooked wine was never included in the prohibition . There is a dispute amongst contemporary Rabbis if the pasteurization process renders the wine as cooked or not , one can be lenient, since this is a doubt on a rabbinic prohibition. Now we will explain the law regarding wine that was poured by a non-religious Jew . If he would open a bottle of wine, the wine is certainly permitted, as was mentioned earlier regarding a non-Jew, all the more so when it is opened by a Jew. If a non-religious Jew pours a cup of wine , if he is the type of Jew who does not fulfill the ritual commandments (bein adam LaMakom), since that is how he was educated and raised from a young age , then the wine does not become prohibited by his pouring or his touch. However, if he declares about himself that he willingly is a non-believer as the result of his investigations and searches , and he publically violates Shabbat, meaning he violates Shabbat even in front of a great person , then he is considered as someone who is a heretic on the entire Torah , and the wine would be prohibited. In any case where there is a doubt as to his status, one may be lenient, as is the case with every doubt on a rabbinic prohibition. A full response in Hebrew with sources can be found Bemareh Habazak volume 7 siman 62: http://eretzhemdah.org/data/uploadedfiles/ebooks/36-sfile.pdf
According to R' Moshe Feinstein this is just a chumrah which does not apply to most non-Shomer Shabbat Jews today.
As summarized by dinonline:
Some authorities rule that one can be lenient concerning the touch of wine by those who violate Shabbos publicly (see Iggros Moshe Orach Chaim 5:37).
The reason for this is that although the Shulchan Aruch (124:8) rules that wine touched by a mumar if forbidden (as emerges from Chulin 72), this is only because of a concern that the person is an idolater. Today’s non-observant Jews are not idolaters, and there is no concern for idolatrous libations, so that the prohibition does not apply.