Why do you have to mend anything with him? If you're right, you're right That's all there is to it. Why fight the fact that you're right?
If you are wrong, cheerfully admit that you are wrong about it, be chill about being wrong and figure out how and why you went wrong.
If you are are right about it, be chill about it, shake everybody's hand and move on.
At least, that's the way people should act.
Your coworker vehemently disagreed with you and now, he's proven vehemently wrong. What goes around comes around. He did it to himself, let him deal with it. Since he invested so much of himself into his disagreement, let him deal by himself, with the personal issues he has created for himself.
Be unapolegetic about being right, but make it clear to everyone that you have moved on and that you are keeping the lines of communication open and of course, your door open.
If you need his cooperation about a specific issue, you talked to him and you're not getting it, then his non-cooperation becomes a job performance issue and you need to raise with your manager the issue that he is not cooperating, or not cooperating in a timely, adequate fashion.
He blew up his side of the bridge with you. Let him rebuild it. It's only when you need him to cooperate with you and you're not getting it that his problem becomes your problem. And if it becomes your problem, make sure that your manager knows that it's their problem, too.