Interesting that he wants his parents involved. It's possible that they said he was grounded and he needed to apologize - but a lot of kids (especially those forced to say "I'm sorry" at age 3 when they weren't!) think an apology is a do-over and implies some obligation to take them back and pretend it didn't happen.
I think forbidding a relationship never works, but I think empowering a teen (especially a girl) to say, "I deserve better than this" is a good thing. It's also not your daughter's obligation to take him back just because he apologized. But if she wants to, you can consider letting him and his parents into your living room (your territory) and just listening to what he says. Don't respond, and tell your husband to contain his body language and eye-rolling. Don't fill the silences. Let the boy do that. Let him squirm.
I think it's important to separate whether they break up or have arguments from what he does on line. It's fine to state your concerns (after he goes through the squirming) - say you think he sounds sincere in his remorse, but you have concerns about his decision to go public with a private argument/dispute, and you wonder what else he will do on line if they argue again. Then stop talking and let him think about it and try to come up with an answer. The less you say, the more he has to talk and the less your daughter will say you are ganging up on him.
I don't know what the parents will say. If they say, "He wasn't raised like this and he knows better," you have to find a diplomatic way of saying, "Well, obviously not since he did it!" Like, "It's difficult to control what our children do on line once they have phones and computers. So what do you suggest? How will things be different in the future?" If they come up with something like, "Well, he knows now and he learned his lesson," you can respond with something like, "What's a good way to ensure that? My concern is that once something is out on the internet, it stays out there forever. Even if someone takes down a post or a tweet, someone else has shared it, forwarded it, or taken a screen shot of it." Again, let everyone else do the talking while you just listen. You have to have a totally neutral tone in this though. It's very hard to not react (either in anger or immediate forgiveness or all-is-well) - you have to be unemotional and kind of blank. That puts the onus on them and lets them find new ways to say things. Based on how agitated or angry they get (or don't), you'll get a good indication of what the dynamics are like.
It's good to let the teens know that their actions say more about them than about the person they bad mouthed. So this boy putting something up about your daughter actually damaged his own reputation as well.
If it happens again, you have to have a good hard talk with your daughter about why she thinks she should put up with this from this boy. What will she do with the next boy? And the next? And the 25 or 35 year old man who belittles her? There's also a lot of pressure on women to do stuff to make men feel better - to take more care of the man's ego than her own. Work on that if you see signs of it.
Forbidding a relationship makes kids sneak around, and once they turn 18, they have even more freedom. You don't want a relationship of defiance going on.