I understand integrity, i really do, but if you start doing this it can lead to nothing but bad things, i.e., coach tells other coaches (which they do), he is still pissed and starts ripping you (in which case you get mad back at him).
Only put yourself in off-court situations in which their is a win-win, in which most cases there aren't.

Allow administration to handle these situations. If we miss a call and a coach has a sincere complain about it the admin will review the tape and get back to the coach on the play. Part of being a being professional is having professional relationships, which can look like friendships but don't entail all the benefits of a true friendship, meaning you only make contact in a business related setting, and talk and do things with a business-like approach. In my mind this would not entail emailing or calling a coach. I do understand, though, that things are different at the HS level, and even some in the college game.

You are kidding with all of this right?

At the HS level If anything they are going to say that you owned up to your mistake and will take you back in a heartbeat because they know you are fair. Don't get me wrong... you can NOT do this very often b/c your credibility for knowing the rules will take a hit. But if you boot a big one I see no problem extending a quick communique acknowledging a mistake. This does not create any kind of improper relationship... give me a break. Coaches are people just like us. If I tell him I screwed up that does not make me his buddy, or unproffesional or anything else nor does it give the appearance. I can promise you that I am as professional as the next when it comes to handling my business as a referee. Sometimes I think we make too much of things and create bigger problems for ourselves... we are unaproachable, afraid to admit when we are wrong, bigger than the game etc. Look at the credit Ed H of the NFL has gotten since he manned up and admitted his error to the coach, the league and the fans. He is top drawer in my book (was before this debacle and is even more so now).

In my college games, I will go back through the assignor if such issues occur because that is how he wants it done. He does not want us directly discussing anything with the coaches without his direction. That being said, he did authorize me to give a coach a call once to discuss a mistake that I did make in a game. Coach thanked me for the phone call and I have been working his games since... and guess what... no one thinks I am his buddy or anything else... In fact, I had to stick him. Guess what, he knew I had to stick him because he knows I am fair and he apologized to me 2 minutes later in the game.

Occasionally I Make A Mistake, And Occasionally There Is A Blue Moon ...
 

I would send an email, containing the apology, to my board secretary, with copies to our president, and assigner. I would ask the secretary to forward the email to the athletic director.

If you know you have kicked a rule contact your rules interpreter and/or supervisor (or the AD if that is who hires you) and let them handle communicating with that coach. If you happen to see the coach again then you can bring it up in the pre-game handshake. Short and simple, "Hey coach, that play I had last game with you, I blew the call". Then keep moving.