This is the most extreme expression of a common canard: fasting in Nisan is inconsistent with the Jewish tradition, so how can you put a day of mourning in the last week of Nisan?
Except, of course, that long before 1953, when the State of Israel established Yom HaShoa, there was a day of fasting in the middle of the final week of Nisan. That dates back to the 40's--not the 1940s, the 740s.
These are the days on which we fast according to the Torah; whoever fasts on them must not eat or drink until evening: on the first of Nisan, Aaron's sons died; on the tenth of Nisan, Miriam died and the Well vanished; on the twenty-sixth, Joshua son of Nun died. (Halakhot Gedolot, ch. 18, p. 232)This 8th-century Gaonic source is echoed by the early prayer books of Amram Gaon (Order of Fasts) and Mahzor Vitri (ch. 271). The Kol Bo (ch. 63) concurs, but records the date of Joshua's death as the 28th. So, somewhere between the 26th and the 28th of Nisan, there was a strong custom to fast; somewhere between the 26th and the 28th of Nisan, we have Yom HaShoa. Yet we can't mark this day--because of the added mourning customs we've tacked on during this period after the Crusades 350 years later?!
Well, maybe these days were once observed, but surely the decisive compendium of Jewish law, the Shulhan Arukh, wouldn't mention--oh, never mind, it's quoted verbatim in OH 580:2.
OK, let's put aside the technical halakhic question (I'm not aware of anyone who actually fasts on this day anyway). Can the yahrtzeit of Joshua shed any light on Yom HaShoa? I believe it can. Quite simply, Joshua (along with Caleb) is the survivor of the worst generational holocaust in Jewish history. Out of about 600,000 able-bodied men who leave Egypt, only two enter the land. This catastrophe is caused by the propaganda and demagoguery of ten men, the Spies, who break the will of the people in the name of preserving the Chosen Race. For forty years, the Israelites must pay the heavy price of this historic mistake. Finally, once they enter the land, as we read on the first day of Passover (Joshua 5:4), "This is the reason Joshua circumcised them: all the people who came out of Egypt, who were males, even all the men of war, died in the wilderness by the way, after they came out of Egypt." Normally, it is the father who circumcises, but the generation that entered Israel was an orphan generation; Joshua had to foster a fatherless people.
Upon entering the Land of Israel, I wonder if there were any great thinkers who opined that this tragedy was necessary, that only a catastrophe as vast as the death of 603,548 men would allow God to give us such a great gift. If so, they would have been immediately recognized as fools. God promised the Land to Israel on their way out of Egypt, and the holocaust in the desert only served to delay it.
So why do people insist on putting Yom HaShoa and Yom HaAtzmaut on two opposite sides of the scales of theodicy? As my rebbe, Rabbi Yehuda Amital was wont to say, if God offered us such a deal, it would be morally repugnant to consider it. The Holocaust must be evaluated on its own terms, not least of which because so many others aside from Jews were slaughtered in it as well. It must be discussed, knowing that it can never be fully explained or understood. Adding it to the list of Jewish tragedies on Tisha beAv, when neither schools nor yeshivot are in session, hardly fits the bill.
I think that the yartzheit of Joshua, who buried more than half a million of his kinsmen, friends and countrymen, is a particularly opportune time. Maybe some other day would be more appropriate. But the vagaries of the Jewish calendar should not be an excuse for ignoring the Holocaust and the profound questions it raises.