R. Moshe Lichtenstein has an excellent article in the most recent issue of Alei Etzion on the religious significance of Israel Independence Day (link). Let me summarize it, at least as I understand it. He has two main points which lead to his conclusion:
I. Secular Zionism
Secular Zionism is a renewal of the Covenant of the Patriarchs (Berit Avot*). This covenant includes connection of the nation to the land of Israel (cf. Gen. 17:7-8; Lev. 26:42). Secular Zionism reestablished a bond of Jewish nationalism and a link to the land of Israel, culminating in a state. While this does not include the Covenant of Sinai (Berit Sinai*), which involves observance of the Torah, it is still a significant renewal of a biblical covenant with God.
Jewish holidays do not commemorate miracles per se, but covenants. Thus, there are miracles that have no holidays (e.g. Jericho) and holidays that have no miracles (e.g. Sukkot*). Chanukah and Purim, in particular, represent renewals of the Berit Sinai.
III. Yom Ha-Atzma'ut
Thus, Israel Independence Day commemorates the renewal of Berit Avot* and, therefore, qualifies as a Jewish holiday: "It is neither the actual establishment of the national entity, nor the claims of "the heels of the Messiah," but rather the renewal of the covenant on the part of the generation of Israel's independence that lies at the heart of this festival."
R. Lichtenstein's evaluation of Secular Zionism is, in my opinion, brilliant and innovative.
* Note that in honor of Israel Independence Day, I am transliterating in Modern Hebrew.