Secular Judaism is an Oxymoron
Judaism is a religion.
"Secular" Judaism, sometimes called "Humanistic" Judaism, sometimes called "Cultural Judaism," is an oxymoron. That is, the adjective "secular" and the noun "Judaism" that it modifies contradict one another. Thus to assert the existence of "Secular Judaism" is like asserting "dry water" or "wet fire."
Judaism is a Religion
The contradiction here lies in the fact that "Judaism" is a religion; it is the body of doctrines and practices that constitute the relationship of the Jewish people with the God with whom this people lives in an everlasting covenant (brit). In fact, in place of the rather modern term "Judaism" (yahadut), it is more accurate in the context of Jewish tradition to speak of Torah.
The Torah is the written and oral constitution of the covenant between God and this people Israel (keneset yisrael). Therefore, the inherently contradictory character of "Secular Judaism" becomes more apparent if one were to say Secular Torah, that is, when secular means "[t]he metaphysical rejection of the transcendent" (as Rebecca Goldstein correctly observes in her otherwise problematic paper).
The way out of this conundrum for some Jewish secularists has been to go back to the Hebrew word usually used to denote "Judaism"--yahadut--and translate it more literally as "Jewishness." This move itself goes back to the term Yiddishkeit (often used, though, by very traditional Jews of East European extraction to designate their intensely religious Judaism).
What this linguistic turn seems to accomplish is to separate traditional Jewish belief from practices that are identifiably Jewish; in other words, to secularize them.
Who Are "Secular Jews" Then?
There are self-styled "secular Jews," for example, who celebrate Passover (which sociologists tell us is the Jewish ritual observed by more Jews today than any other) not as God's redemption of Israel from Egyptian slavery but, rather, as the autonomous acquisition of freedom and sovereignty by the Jewish people in the founding event of their "civilization" (a term favored by the equivocally secularist Mordecai Kaplan, and which is as vapid as the term "values" employed by the unequivocally secularist Rebecca Goldstein).
Nevertheless, the question is not only how much this deconstruction of traditional Jewish practice corresponds to the Jewish tradition as a whole (whose two constituting documents are the Bible and the Talmud), but also how coherent this kind of Judaism or "Jewishness" really is (a "philosophical incoherence" Goldstein herself honestly admits). So, if Passover is a cultural event, which does not require religious commitment to be properly celebrated, why does one even have to be "Jewish" to celebrate it, that is, when one's very connection to the Jewish people does not have the rigorous consistency of religious norms to define it?
Indeed, many of us know or know of non-Jews (usually Christians) who claim to be celebrating Passover. (That is why, by the way, I don't like Passover celebrations conducted by non-Jews among themselves, even though I see no reason why a non-Jew cannot be a welcome guest at the Seder of a Jewish friend.)
I have no objection to calling something like Passover a cultural celebration, that is, if one keeps in mind that the words "culture" and "cult" (in the sense of religious worship) come from the same root. And, in fact, I know of no historic culture, as distinct from ideologies such as "secularism," in which religion (as defined above) is not central. That is why so-called "cultural Jews" not only misunderstand Judaism or even "Jewishness," they also misunderstand what culture is essentially even more so.
The same type of misunderstanding can be found among Jews who believe and practice "Humanistic Judaism," since the uniqueness of being human, for Judaism, is that humans are created in the image of God, which for me (following my late revered teacher, Abraham Joshua Heschel) means that humans are the objects of unique divine concern, that they alone of all God's creatures can be the recipients of revelation (Avot 3.18).
To be sure, one need not be a kabbalist, for whom Judaism (the Torah) is the only reality and that reality is nothing but God, to see that a Judaism or Jewishness devoid of a strong God-connection is bound for philosophical incoherence and historical oblivion.