Why was Yom HaShoah, the memorial day for the six million Jews who perished in the Holocaust, not established as a fast day? That would be consistent as we fast on other days when we commemorate past mass destruction and murder.
This is because Yom HaShoah is held on the 27th of Nissan, and it is halachically forbidden to establish a fast during the month of Nissan. (the fast of the firstborn does not fall under this prohibition).
In our kinnos (kinnah 25, about the first crusade) the paytan writes: "שימו נא על לבבכם מספד מר לקשרה, כי שקולה הריגתם להתאבל ולהתעפרה, כשרפת בית אלקנו האולם והבירה, וכי אין להוסיף מועד שבר ותבערה, ואין להקדים זולתי לאחרה, תחת כן היום לויתי אעוררה"
It was for this reason certain gedolim in Europe didn't want to establish a new day for Churban Europa. Another source for this same idea can be found in Rashi on Divrei Hayamim 2 (35:25).
Similar to the already answered answers, in the sefer הדור והתקופה page 65 it is brought down that when the Israeli Chief Rabbi, Rabbi Hertzog came to speak to Rabbi Yitzchok Zev Soloveitchik, the Brisker Rav, about setting a fast day for the horrific events of the Shoah, the Brisker Rav responded by quoting Rabbeinu Klonimus' Kinna "אין להוסיף מועד שבר". At that time the Eida Chareidis didn't join the Rabbanut in their call to establish a fast day, and it seems that that was the reason for the Rabbanut to not try and create a fast day.
Additionally, in the book פאר הדור ח"ג עמוד קכ"ד it is brought that Rabbi Binyamin Mendelsohn asked the חזון איש why we don't establish a fast day for the Shoah, even though the ט"ז established one for גזירות ת"ח ות"ט. The חזון איש responded that firstly, it is not clear in the ט"ז and additionally, we are not on the מדריגה of the ט"ז to copy him.
For further reference see שיחות בהלכה (Brooklyn,NY 2022) pages 81-83.
This Sefer also mentions the halachic issue of setting such a date in Nissan.
As I heard in a lecture from Rabbi JJ Schacter:
Rav Moshe Feinstein explains that all national-level tragedies are mourned on Tisha B'Av, and the Holocaust was a national-level tragedy. (He points out that had Hitler succeeded, he would have gotten rid of Jews all over the world.) While some communities do have minor fast days for more localized events, those are smaller.
Yom HaShoah was established by the State of Israel as "Yom HaShoah Ve-haGevurah" -- "the Holocaust and courage." The government chose the date of the Warsaw Ghetto Uprising because it symbolized Jews fighting back. It wasn't created as a religious day per se, nor by religious authorities.