I don't remember the original incident...I assume things got physical between you two? And this happened in New Jersey?
You don't have to worry about being arrested, even if it was physical, as long as you didn't cause him grievous bodily harm or something else that would have made it a felony.
First, in New Jersey, the statute of limitations for misdemeanors is one year. If you didn't cause him serious physical injury (like put him in a hospital or cause him harm to the point where his life was actually in danger), New Jersey considers it simple assault which is a "disorderly persons offense," which means a misdemeanor.
Second, even if there wasn't a breach of statute of limitations and there was a warrant out for your arrest (which at this point, I'm sure there isn't), if it was for a misdemeanor they aren't going to extradite from out of state. Basically, if you have misdemeanor warrant, it's HIGHLY unlikely and extremely out of the ordinary for any police department to bring you in on it unless they catch you or another police department catches you within 50 or so miles of the venue where the warrant was issued. It usually is just seen as not worth the time and money to transport someone that far for something and hold them for that long unless it's a more serious charge.
That's all kind of a moot point though because like I said, I'm pretty positive you don't have a warrant anywhere.
Bottom line...as far as being arrested goes, I can almost guarantee you can put your mind to rest about that.
As far as apologizing to your cousin and making amends...that's only for you to decide. Not your family, not your friends..you. You need to weight out if it's more important to you to stand on whatever principle is keeping you from extending the olive branch or if it's more important to compromise for the sake of peace in your family. Maybe you should talk to your father more and try to get him or someone else to see if your cousin will make the first move toward amends?
Wait, I thought "everything is legal in New Jersey"?
(Hamilton reference, because not everyone else quotes from it on the daily.)