I was surprised to see the extent to which this seemingly innocuous question is actually a hot-button issue. After all, we come across essential points such as whether we should have waited for the Messiah to bring us to Israel or how we should view the celebration of a secular state. Hallel, which is Psalms 113-118, is not something we can just recite capriciously. Even the Gemara speaks harshly of those who recite it everyday (Shabbat 118b). Hallel is only reserved for special occasions. As such, we need to be prudent in approaching such a halachic question.
As to whether we should even have a Yom Ha’atzmaut, I’ll be brief. Most of those who don’t give any credence to Yom Ha’atzmaut are certain members of Haredi Orthodoxy. Short of that, it is universally accepted as a modern holiday. Even the Chief Rabbinate has declared it a holiday, as well as helped in creating new liturgy expressing Zionist sentiments on this day. Sufficeth to say, with its overreaching acceptance, we can, at the very least, treat it as a minhag.
Now we come to the question of whether we can recite Hallel for Yom Ha’atzmaut. We only recite Hallel under certain conditions. The first one is that an open miracle took place. There are rabbis who argue against reciting Hallel because they believe that a blatant miracle did not occur. I beg to differ. Israel is surrounded by the antagonistic nations of Syria, Egypt, Lebanon, Iraq, Yemen, and Saudi Arabia, all of which are ready to pounce on Israel and eradicate the Jews from the land. The Jews were outnumbered, had inferior weaponry, and a comparably small amount of assets [compared to those nations surrounding them]. Geographically speaking, Israel’s location makes it highly vulnerable to invasion. If one calculated the odds, there was no way that Israel could win, but lo and behold, it happened. It is such a blatant miracle that when West Point was asked why they don’t teach this war in their military school, their response was “we don’t teach miracles.” Since there is no geo-political or militaristic reason for Israel's victory, the only explanation is Divine Providence.
R. Azulai opines that an event that represents a step in the rebuilding of the Beit HaMikdash (i.e., the creation of the modern state of Israel) warrants reciting Hallel [with a blessing]. Complaints against this notion are that Israel’s shortcomings negate the recitation of Hallel because Yom Ha'atzmaut is not fully redemptive. Rav Ovadia Hadaya, for instance, rules that Hallel cannot be recited because of Israel’s unstable security. Even Chazon Ish (Letters of the Chazon Ish, #97) stated that Israel’s spiritual shortcomings, i.e., the insufficient amount of Torah observance, doesn’t allow us to institute a new practice such as this. However, many rabbis with notoriety, such as R. Aharon Soloveitchik and R. Meshulem Roth, disagree. When one looks at Pesach or Chanukah, both the Exodus and the militaristic gains were both the beginning of redemption, yet we say Hallel for both. Even according to R. Ya'akov of Lisa, one can establish a holiday for the entire Jewish People to commemorate an event that is considered an act of redemption.
Another issue is whether the miracle happened to the entirety of the Jewish people. With Pesach or Chanukah, the miracles occurred to all Jews. In this instance, we see that the War of Independence did not affect all Jews. Normally, one would think that this would disqualify us from reading Hallel. However, the caveat here is expressed by Meiri’s ruling, based on Pesachim 117a (the Talmudic source of reciting Hallel), which states that if a miracle happens to an individual or a portion of the Jewish people, Hallel can be recited without a bracha.
Conclusion: Although you can easily find a rabbi on either side of the argument, a disinterested look at it provides ample evidence to show that Hallel can be recited. However, since this miracle did not occur to the Jewish people as a whole, a bracha technically cannot be recited. After much contemplation, I have decided to go with reciting Hallel without the bracha. Whatever your approach to this modern holiday is, I hope that you can find a way to use Yom Ha’atzmaut to enhance your connection to the state of Israel, as well as G-d.