The source for the obligation of the recitation of Hallel is in Pesachim 117a, “The Rabbis taught, ‘Who instituted this Hallel? The prophets legislated for the Jewish people that this Hallel should be recited at each designated season and on any trouble that should not befall them, and that when they are redeemed, they should recite it upon their redemption.’”
There is no doubt that the situation in which the Jewish state found itself just prior to the Six Day War and the incredible redemption within six days from it (that amazed the entire world) satisfies the criteria that our Sages established in these matters.
There yet remains to be determined if the obligation to say Hallel is just a one-time concept or a reoccurring one. The Netzivof Volozhin is his work Ha’amek She’elain Parshat VaYishlach, Siman 26 wrote that this is a one-time obligation. However, from the Chatam Sofer it is clear that the requirement is an annual one.
The later authorities commented that the outstanding poskim (halachic decisors) including the Rif, Rambam, and Rosh do not mention this halacha. Nevertheless it is difficult to explain why we would not accept this halacha since it is a definitive ruling?!
There are many who argue that this halacha only refers to a potential tragedy that affects the entire Jewish people, but not for an individual or city. Of the opposing view are those who consider a “Tzara”that threatens the Jewish people who reside in the Land of Israel as tantamount to a threat to all of Israel – as the matter appears in regards to Chanuka.
The last point one needs to analyze is that we do not find a Takanato say Hallel on the liberation of Yerushalayim from our enemies?! Therefore there are those who argue that it would be more proper to affix the day of Hallel on either the second day of Sivan that was the last day of fighting or the third day of Sivan that commemorates the end of combat (similar to Purim that was established on “the day they rested from their enemies.”)