The problem is that by even mentioning it, he is shaming her. Just even a small polite mention. She will feell judged and shamed. Because it is reinforced by all the cultural forces you mentioned. And it sounds like she is already aware of his “issue” with her armpit hair to have gone through all these explanations with him--she is already on the defensive, about her own body due to how he feels about it. So his reaction has already been noted with all its western kyriarchical power.

So his choice here is to keep pushing his kyriarchical power on her, with the end point of breaking up if she does not conform to his demand, or work on changing his own reactions by thinking more about the issues of power involved. Sure he has a right to break up over it, but that will still be an exertion of kyriarchical power over her she will carry with her. I could only advise him to work on himself or go on being an abusive jerk. It’s not a simple matter of aesthetic taste.