From what I've seen (primarily on Stack Exchange), there are a few relevant factors:
If you've just broken up a rapid-fire back-and-forth, where responses are coming in close to real time, then a forced time-out for a few hours can sometimes calm it down. Freeze comments on the post, disable the chat room where it's happening, or otherwise block them from continuing the fight where it is occurring. They might take it elsewhere, but I've seen freezing a chat room for even 10 minutes stop this sort of thing in its tracks, so it's worth a try.
On the other hand, if this has been going on for days, with the mean time between comments being measured in hours rather than seconds, this won't make a difference.
Acute or chronic?
If two normally-reasonable users are blowing up at each other over one particular issue, you probably want to handle that differently than if two people keep going at each other over and over. Assuming I value both of the users involved, for an acute issue I'll make "less formal" contact -- a private chat, for example -- with each, asking the user to calmly explain to me what the problem is. "Never mind that other guy you're arguing with; he's not part of this conversation. I'm a mod and I'm here to help; talk to me." The goal is two-fold: (1) maybe you'll better understand the issue, but (2) more importantly, each will feel like he's finally being heard by someone who matters. Even if you don't do anything about it, you've given each a sounding board.
For a chronic problem, or if the less-formal approach didn't diffuse an acute problem, it's time to reach for a more-formal warning, something "on the record". With luck you already have terms of service, or a code of conduct, or other rules that cover how users are and are not permitted to behave on the site. With luck, those rules include something along the lines of "be civil; disagreeing is fine but nastiness is not". Quote from (and link to) this in your warnings, say that you value the user's participation (if you do) but this behavior must change, and explain that the next incident will lead to a temporary restriction on posting ability.
But I made it worse!
Yeah, maybe you did, and in your communications with the users involved you can acknowledge that and apologize for it if you feel it appropriate. But that doesn't make their behavior ok, and you may need to say that explicitly. Your duty is to the community first and foremost, and that means shutting down nastiness that makes your community an unpleasant place to be.
Is help available?
If you are not your community's only moderator, and you feel that you've accidentally poured oil on the flames, then getting a different moderator to step in can help. If both of them are mad at you, and you have the option to not be the next person who contacts them about it, having someone else do it may calm things down more quickly.