There are two major possibilities for the user's behavior:
- They don't understand that Stack Exchange isn't a forum, and they don't totally understand how comments should be properly used.
- They know how to use comments but don't care.
It's always nicer to assume that the first reason is to blame - especially if the user doesn't have much experience on Stack Exchange as a whole. I like to assume good intent when possible. It's the more optimistic solution.
If you want to work on this assumption, you can direct the user to chat and try to educate them a bit about comments. Maybe you can give them some links (e.g. 1, 2) for further reading. This is one of those things chat can be great for: a third place to teach other users about the site and how it works. It's one of the advantages of having a third place.
The user should be able to use chat (needs 20 rep) if they have the ability to comment everywhere (needs 50 rep). If they don't have 20 rep on any Stack Exchange site, then you might not have the chat option. At that point, it's possible that you could condense the information into a comment. But chat really provides you with more flexibility here.
All that said, there's a chance that the behavior is really caused by the second reason, and that's when you hit trouble. The user might have a past you don't know about, or might have a long-standing pattern of this specific behavior. Maybe they don't, maybe they do. You don't really have a good way to tell.
If you think this is the case - or if the first approach fails - you might want to notify a mod via a flag (or a ping in chat, but flags are more private). Just say something along the lines of
Hey, [User X] has been really argumentative/is abusing comments. Can you maybe talk to them?
Some things to avoid:
- Don't actively go out looking for users that have been problematic. Deal with issues as they come up, and don't take it upon yourself to deal with all the cases (not saying that you'd be inclined to do so).
- If you don't think engaging directly is working, don't be afraid to step back and let someone else do it. Likewise, if you see someone trying to help a user and having problems, don't be afraid to step in, if need be.
- Definitely don't get into arguments in comments over this (actually, avoid comments if at all possible). It risks defeating your point about not arguing in comments.
TL;DR: Take this on a case-by-case basis for now.