So I'm feeling much, much less upset about the awful student thing. Thank you, everyone, for your ideas and support. I've now spoken with my department chair, division chair, and dean; they all had slightly different approaches to the situation, so I'm still deciding what to do.
I don't think that I'm going to move to kick the student out of my class, at this point (if he does anything else, though, that might happen). That probably wouldn't be in his or my best interests, since he has a lot of friends in the class, and he might end up just feeling stifled for having unpopular views. This is not my primary concern, but it does figure into my calculations.
The best (realistic) outcome of this whole situation would be if he were to learn that saying and writing such things is totally unacceptable and wouldn't do anything else to press these points in class, ever. Ideally, of course, he would also rethink his views of women. The latter is unlikely to happen anytime soon, however, and my kicking him out of the class is very unlikely to reform him at all--although, again, that is definitely a secondary or even tertiary consideration.
So. What to do? Well, two options. (1) I can have a formal meeting with him and another campus authority in which we take the "This is hate speech and will not be tolerated" approach. In this scenario, I wield my power like a mighty ax and emphasize the awfulness of what he's saying, how it makes people uncomfortable, etc. Or (2) I can meet with him one-on-one and discuss how what he's written is not a good example of persuasive writing, the importance of taking one's audience into account, and how such vitriol is precisely what alienated him from the article he's responding to in the first place. I could then move into talking about the effects of using sexist language in class, and how that's unacceptable and will not be tolerated in the future.
In both scenarios, he re-writes the paper.
At the moment I'm actually leaning towards no. 2, with the intention of watching his future papers and comments very closely and also of talking to his coach to make sure that this isn't part of a visible pattern of behavior (because obviously it's part of some larger pattern, unless he's really just trying to upset me, which I doubt). The upside of no. 2 is that he might actually learn something, and the fallout for me will be less: like I say, he has a lot of friends in that class. And if I present it as calmly and professionally as I intend to, then he'll be denied any satisfaction that he might have hoped to derive from upsetting me--if that was, in fact, his intention.
The downside is that I'm not sure I feel that this is a strong enough course of action. It was the one that the dean recommended, however, and she made some good arguments in its favor. (She read the paper, so she knows what we're dealing with.)
It would also be the easiest for me to carry out, frankly.
Anyway. At this point I'm calm, my rage has burnt itself out (thank God), and I'm ready to just settle this. I feel okay now, though. I was having some serious "I can't do this" feelings on Saturday night, and now those have dissipated; classes today were perfectly acceptable, and I'm not getting down on myself about any of this. I mean, one kind of nice thing is that there's no conceivable way I can blame myself for any aspect of this situation; I know that's sort of an odd thing to be comforted by, but it is a relief. My teaching is not at issue here. It's some other big scary thing instead.
Yikes. It's comforting in a rather selfish way, I suppose.