Question: How is it possible to establish a new holiday on Yom Ha-Atzmaut?
Answer: Rabbi Moshe Sofer of Pressburg ruled that there is a Torah obligation to establish a holiday for every deliverance of the Nation of Israel (Shut Chatam Sofer Orach Chaim #191). And yet perhaps there is a question of "Bal Tosif" (a prohibition against adding commandments), as the Ramban wrote that Yerovam ben Navat transgressed this prohibition by creating a holiday. The Chatam Sofer responds: a transgression only occurs when we create a holiday on the same level as a Torah holiday, i.e. with a prohibition against work and similar things. Otherwise, there is no prohibition of "Bal Tosif." The proof is Purim and Chanukah.
There are those who ask: Why wasn’t Yom Ha-Atzmaut established when Yehoshua conquered the Land of Israel or when Ezra and Nechemiah returned to the Land? The "Yom Ha-Atzamaut" after Yehoshua's conquest is Pesach. This holiday does not only celebrate the Exodus from Egypt, but also our arrival in the Land, which is the whole point of the Exodus. This is explained by the four phrases of the Redemption (Shemot 6:6-8): "And I will remove, and I will save, and I will redeem, and I will take," which is followed by the fifth phrase: "And I will bring you [to the Land of Israel]." Regarding the return to the Land by Ezra and Nechemiah, the answer is simple: They did not celebrate Yom Ha-Atzmaut because they did not have independence. They only achieved independence on Chanukah. Their "Yom Ha-Atzmaut" is Chanukah.
An additional question arises: Isn't it stated in the Gemara in Rosh Hashanah (18b) that after "Megillat Ta’anit was abolished" one cannot add any holidays? Megillat Ta’anit is a work of the Men of the Great Assembly that preceded the Mishnah (and it is mentioned in Mishnah Ta’anit 2:8). It enumerates approximately forty holidays which were established during the Second Temple Period, some for military victories and others for spiritual victories. All of these holidays, except for Purim and Chanukah, were abolished after the destruction of the Second Temple, so who are we to add new ones (Rosh Hashanah ibid.)? Furthermore, since we are in a general spirit of sadness and mourning over the destruction of the Temple, and it is forbidden for us to fill our mouths with laughter (Berachot 31a), how then can we rejoice on new holidays? Purim and Chanukah were permitted only because they contained commandments, and it is impossible to abolish commandments (Rosh Hashanah ibid.). There are Rabbis, however, who argue persuasively that holidays continued to be established even after the destruction of the Temple. Rabbi Moshe Sofer of Pressburg held a middle position: The prohibition applies only to establishing a holiday for the entire Jewish People, but an individual may establish a holiday for himself or an entire community. In fact, the Rambam established a holiday for himself on the day he arrived in the Land of Israel (brought at the end of Sefer Charedim of Rabbi Eleazar Azkari), and so too many communities established a holiday for themselves commemorating the day of their deliverance. There are approximately two hundred special "Purims," such as "Purim Borghel" celebrated by the Jews of Tripoli (29th of Tevet, for being saved in 5554 from destruction during occupation by Borghel Pasha of Turkey), "Frankfurt Purim" (20th of Adar, for the readmitting of expelled Jews being to the town in 5376, and the execution of the Chief Jew-baiter, Fettmilch), "Lepanto (Greece) Purim" (11th of Tevet, for Jews saved from destruction during the Turkish War in 5460), "Tiveria Purim" (7th of Elul, for Jews saved from danger of war in 5503), etc. etc.
The Chatam Sofer, on the other hand, permitted establishing a holiday for the entire Jewish People, with the condition that the deliverance was from death to life. His proof is from the words of our Sages: Purim should certainly be a holiday based on Pesach, since Pesach was a deliverance from slavery to freedom. Should we not celebrate a holiday all the more so for Purim which was a deliverance from death to life (Megillah 14a)? He claims that in exile there is no possibility of being delivered merely from slavery to freedom, since in exile we are slaves to the nations of the world. Any deliverance, therefore, could only be from death to life. We may thus establish a new holiday. In truth, on Yom Ha-Atzmaut in the Land of Israel there was a double-deliverance: from slavery to freedom, from the British authority to Jewish authority, and from death to life, from all of the Arab Nations who tried to destroy us at the time of the War of Independence.
There are those who claim that before the War of Independence we were not really slaves and there was therefore no deliverance to freedom. This is incorrect - we were slaves in the full meaning of the word, since the Talmud explains that we do not recite Hallel on Purim, since it is written in Hallel "Praise, servants of Hashem" and even after the miracle of Purim we were not servants of Hashem, but still servants of Achashverosh (Megillah ibid.). Before the establishment of the State, we were slaves to the British authorities. And now – through Hashem's kindness – we are a free Nation in our Land.