You can’t fix the problem, whatever it is.
So here’s what I would suggest (because I’ve done this):
1. Invite them both to a weekend breakfast, brunch or lunch (and pick up the tab). Don’t surprise either one of them: they should both know ahead of time that the other will be there. Be the adult at the table and let them know that they’re both important to you.
2. Ask them to be civil with each other and see if there’s a way to resolve the issue between them. That is, the “proximate issue” – what was said in bantering that got out of hand recently on Facebook, and then “give thought to” whatever was bugging them before that. (Don’t get so ambitious that over one meal you’ll put things perfectly right between them.)
3. Thank them for coming and for at least attempting to keep you from having to choose sides in the short term or long term – no one is going to “win” if they make you do that, and they should recognize that fact.
4. Either you leave first or let them leave together (so you won’t be at the table talking to one about the other after he or she leaves) and leave a nice tip for the server.
Be cool, be in charge, and stop the conversation if it gets out of hand, off topic or accusatory.
I did it once as the commissioner of a youth recreational soccer league, mediating between two coaches who wanted to kill each other. At the end of the meeting they just “didn’t like each other”, but they didn’t quit the league because the other stayed, and they agreed to be civil in front of the kids. I counted it as a win.