I don't know what the core issue you were fighting about was or who was legitimately in the wrong, but if you broke off an engagement, moved four hours away and never said goodbye, it had to be a BIG blow up. (This was no way to start a marriage, so it's a mixed blessing that it happened before the wedding rather than after.) This huge issue apparently never had closure for you if you're still thinking about him. However, he rebounded the next night with another woman and as emotional punishment to you now won't return any of your messages.
If you truly owe him an apology, you can write him a letter since he's not in the mindframe to hear your voice. You must have his address or the address of a friend, neighbor, or relative of his who can deliver this letter to him. Mark on the outside "To be opened by addressee only." Hopefully that will help ensure that a roommate or other person doesn't open his mail. Say what you need to say and give your contact information, if you want to. Tell him your purpose is to apologize. Less is more so don't ramble. Just own whatever you need to say sorry for.
He has told your sister that he is a loser and you deserve better. He doesn't have a phone and doesn't seem financially stable. He's rebuffed your attempts to contact him and immediately rebounded with another woman. I don't know what the backstory is there, but think long and hard about whether he's right for you. Is he the loser he claims to be? Apologizing is very different from reuniting. Be very clear in advance what you want and why so that you can be clear in what you communicate. Otherwise, you may wind up reliving -- and probably repeating -- the past.