Damore is fired for writing a carefully argued statistical paper.
Damore wrote a set of opinions and associated them with a random selection of what he asserted were facts. However, there are few actual statistics in his paper (there's one about 93% of work-related deaths being of males and another asserting that 95% of humanities and social sciences lean left) and none is actually relevant to his argument (workplace death is not a common problem in IT and he isn't suggesting that HR departments are entirely staffed by social science graduates) .He makes a number of assertions about male vs female characteristics but fails entirely to correlate those characteristics with the tasks staff are required to perform, so they're irrelevant too, regardless of whether they may be true or false. He also has a few links to disputed evidence of bias against men. The only statistical argument I can detect in it is that correlation actually is causation as that seems to be how he explains the present gender balance.
So no, it wasn't anything like a carefully-argued statistical paper and would have been a fail grade if it had been an assignment.
But, I agree that in an "employment at will" situation, I can't seen why "Fuck you" man wasn't fired.