Completely ignoring the sexist aspect of the word, is using "he" as a gender neutral pronoun grammatically correct or incorrect? I'm well aware that using "he" may come off as sexist or politically incorrect and that it's better to use "they" but grammatically is it correct?
According to the Cambridge Grammar, he was the sex-neutral pronoun for quite a while. However, eventually, when equality between women and men became prominent, the use of he became controversial because "[it] represents one of the most obvious and central cases of sexism in the English language..."
So people began using he or she instead of he as a sex-neutral coordination to avoid that sexist bias. But in informal conversation, it says, using it repeatedly can sound clumsy or clunky.
They as a sex-neutral pronoun with a singular antecedent actually dates back to Middle English. Some people use that as an argument for the use. Others argue that they is supposed to be plural, so it shouldn't be used as a singular, sex-neutral pronoun. But even still, they grows in use, and the book seems to accept it as the pronoun.
Of course, that leads to the obvious question of, "Themselves or themself?" It doesn't say that it's specifically wrong, just that it will gain further acceptance as the use of they becomes more common. And it's been thirteen years since the book was written, so I'd like to think it's quite a bit better accepted.
TL;DR: If you don't want to use they because people might object or you don't like themself, use he or she. I personally prefer they.