Was Eisav Jewish? Eisav was born to the same father and mother as Yaakov. It may be possible to say that Yaakov was not Jewish either as it was before Matan Torah, however if Yaakov was Jewish, then was Eisav also?
The Gemara (Kiddushin 18a, top) calls Eisav ישראל מומר - an apostate Jew.
ובקידושין (י"ח א) איתא דעשו ישראל מומר, והיינו דקודם שיצא לתרבות רעה היה עליו שם ישראל ונימול לשמונה אלא שנשתמד, ואף דבישראל גם מומר אף על פי שחטא ישראל הוא, זהו רק בזרע יעקב שכבר נפקע חבל הנחלה בשלושה חוטין דאין יכול לינתק עוד, אבל בזרע יצחק אף דהיה נקרא תחילה בשם ישראל משנשתמד ויצא לתרבות רעה יצא מכלל ישראל
In Kiddushin (18a) it says that Eisav is a ישראל מומר. This means that - before he went off on an evil path - he could still be considered a member of "Yisrael" who had been circumcised at eight days; but he apostasized [and lost this distinction]. Now even though among Jews the rule is that even an apostate, although he has sinned, is still considered "Yisrael" - this applies only to the descendants of Yaakov, since the "cord of inheritance" has already been twined from three strands [the three Patriarchs] and can never be severed. But with Yitzchak's children [this did not apply]: he was indeed at first called by the term "Yisrael," but once he apostasized and went off to evil behavior, he is no longer considered "Yisrael."
The Lubavitcher Rebbe explains the difference between Eisav and Yishmael.
Even though Eisav was evil throughout his whole life and wanted to stop Yaakov from being buried in the Mearas Hamachpeila, Eisav is considered a Jew (though a Mumar, as in Alex's answer).
Yishmoel, though, did tshuva. Yet he was not considered a Jew. Moreover, we see that Eisav, who was wicked, merited to see that his head was buried in the Mearas Hamachpeila. Yishmoel didn't.
The difference between the two is the difference between their fathers. Yishmael's father was Avraham, who functioned in a top-down (chesed) way. His function was to illuminate the world. However, illuminating the world doesn't change it. Therefore, Avraham was able to change Yishmael, but only as long as Yishmael was subject to Avraham's influence. Once the influence ended, he went back to his natural (evil) nature.
Eisav's father was Yitzchak. Yitzchak worked to perfect himself, to reveal within himself the G-dly point. Therefore, in a revealed way Eisav was evil until the end. Yet in his essence (his "head"), he remained connected to Hashem.
First of all, Eisav could not have been Jewish by any standard definition we use today. If you insist on the word "Jewish" then he was not a descendant of Yehuda and therefore not a Yehudi. If you go with "b'nei Yisrael" well he wasn't a son of Yisrael/Yaakov. Chazal and Rashi note that the wording of Torah as to how the zera kodesh is transmitted make it clear that only Yaakov's descendants were entirely part of God's nation. Eisav is no more Jewish than Yishmael or the B'nei Keturah.