[CFrance]

It looks like the opinion is 50/50 about apologizing. I based my opinion on the assumption that you already apologized when the incident occurred, and the neighbor didn't accept it. In which case I still think you've done enough and can just be as friendly as possible from here on out, keeping your dogs on leashes and NOT letting them stray onto anyone's lawn while walking them.

That's a big-ticket item here. Have your dogs do their business on your lawn before taking them out for walks, especially since it seems like you have landed in one of those neighborhoods whose property owners are touchy about that.

I hope things will settle down from here on out.