Have you ever worked in generalist tech support? One of the first things you learn is how to draw a line when the situation is beyond mutual respect, because nothing good can happen there and you have already lost.
A lot of people like to blow off steam by abusing a random voice on the phone, so you are entitled to quickly apologize and cut them off. Same applies (or should apply, in respectable support shops) in other situations the support worker would deem socially unacceptable. So yeah, you can fire the customer, your mental health comes first.
Btw, the most depressing call I've ever had was with this guy I had helped for weeks, solving his "enterprise" problems. At the time I worked with a female colleague, and we would share customers and tickets. Eventually she left, so when I broke the news to this guy, he went "oh, i'm sorry to hear that. To be honest, I preferred her; I always prefer ladies on the phone". That was humiliating beyond belief. When I told her a few years later (since we kept in touch), she was absolutely flattered. Because sexism is not that bad when it benefits you, so to speak.