The determination of Jewish identity is a complicated matter. The answer to such a question, therefore, would depend on the perspective from which one asks. If identity is to be defined by the individual him/herself then external sources become irrelevant -- people are who they say they are. However, most would agree that identity is not so simple, a person's identity is also determined by the people around them -- by their community and the norms of the culture of which they are a part.
According to traditional Jewish law, a person is a Jew if they were born to a Jewish mother or they converted to Judaism according to Jewish law. From a purely legal perspective, therefore, belief in God has no bearing on one's status as a Jew. Many legal perspectives even hold that one who converts out of Judaism is still a Jew, despite their acceptance of other faith principles. Further, it is clear that early Rabbinic Judaism de-emphasized dogmas, or specific faith assertions. One even finds the statement in the Midrash (ancient rabbinic commentary on the Bible) in reaction to the following quote from the book of Jeremiah, which states: "'[They] have forsaken Me and have not kept my Torah'." --'If only they had forsaken Me and kept my Torah.'" This statement implies that doing mitzvot, following the commandments, whatever the rationale was central, while belief in God was less so.
Yet, in truth, as a monotheistic faith a belief in God on some level is certainly understood. Almost all of our customs and laws in some way have theological import and meaning. As a Conservative rabbi, I certainly hold that a belief in God is an extremely important part of any Jewish identity. However, Judaism has rarely required one to define God, leaving room for a wide variety of beliefs about God: God as the mover of history, God as the unity of the universe, God as the spirit inside of all living beings, etc Further, while I certainly think that more people, on some level, believe in God than would say so when asked, I also hold that while a belief in God is certainly central to Judaism, a lack of belief in God does not disqualify one from being a Jew.
There are two ways to be a Jew, to be born one or to convert into Judaism. If you're born a Jew, you are a Jew your whole life, no matter what you do. The principle, articulated by Rashi in a responsum, is "Yisrael af al pi she-hata, Yisrael hu, A Jew, even if s/he sins, is a Jew." That being said, there are Jews who are more or less faithful to their religion, and the community might respond accordingly. In the medieval period, if a Jew converted to Christianity, the family sat shiva, and if that Jew returned, there was a symbolic conversion ceremony performed. We today look at apostasy somewhat differently, but the basic principle still stands-- a Jew who has "converted" out is certainly less connected to Judaism than others, but is still a Jew.
The same would be true for a Jew who does not believe in God-- genetically and by birth, that person will be Jewish their whole lives. However, since the relationship with God sits so centrally to the Jewish religion, that person's Judaism is highly attenuated, at a much lower level than someone who does believe in God and works on building that relationship.
Conversion to Judaism, incidentally, is a religious process and, at least in Orthodox circles but I would think in Conservative circles as well, is impossible without agreeing to faith in God. A convert who did not believe in God at the time of conversion would be an invalid convert, and therefore not Jewish.
My colleagues have answered this question very well, so I don't have too much to add, except for my agreement.
Since it was brought up, I know that there are some Reform Rabbis who will convert a person into Judaism, regardless of what they do or don't believe. Others would say that a person must believe in God, although what it means to believe in God are open to an incredibly wide range of interpretations (most people who say they are atheists are actually agnostic - they just don't believe in a particular image of God which they had been given as authoritative).
As one of the other respondents said, Judaism certainly affirms the existence and oneness of God, even though it never really demands belief, per se. Or, in the words of one of my Rabbis, "what makes you think God cares if you believe in God or not?"
Copyright 2014 all rights reserved. Jewish Values Online
N O T I C E
THE VIEWS EXPRESSED IN ANSWERS PROVIDED HEREIN ARE THOSE OF THE INDIVIDUAL JVO PANEL MEMBERS, AND DO NOT
NECESSARILY REFLECT OR REPRESENT THE VIEWS OF THE ORTHODOX, CONSERVATIVE OR REFORM MOVEMENTS, RESPECTIVELY.