First of all, I think that if the observer really did yell at you to "SHUT UP" in front of the coaches, players and officials then he was unbelievably unprofessional. Having said that, you probably should have just "SHUT UP" in the situation, especially if he's telling you the same thing for the 3rd time. When you receive an observation/critique, if the advice makes sense to you and you think that you can use it, then put it into practice. If the advice doesn't make sense to you, then just forget about it. Don't try to explain yourself to the observer. You never ever ever want to be perceived as a guy who makes excuses, a "yeah, but. . ." guy. That will KILL you.
Second, I once heard on a radio psychologist's show (Dr. Laura, Dr. Joy, Dr. Ruth -- I can't remember which one

) that adults put good things in a letter; but say bad things face-to-face. If you're not brave enough to say something to a person's face, then it's probably not important enough to say at all. That made a lot of sense to me. So. . . if I were in your place, I would not send the letter that you've written.
Third, I'm going to disagree with something a lot of people have suggested. If I were in your place, there is no WAY I would apologize to this guy. He was the jerk; he embarrassed
you. He would not get an apology from me. I would, however, thank him for his time. (It was a sacrifice on his part, even if the results weren't the best.) And I do like the idea of saying that I'd learned how to keep quiet. But an apology? Maybe it's just me, but no way.
Finally, since Dan shared a story, I'll share one too. Dan was even there, so he can verify it. I attended camp this summer and was scheduled for whatever game was next. As the crew from the previous game came off the court, the observer made a big deal about positioning in a certain situation. A
big deal. A very very big deal. So of course, about 2 minutes into my game, I make the very mistake that he'd talked about. The great thing (so I thought) was that I realized it and slid back to the proper position before the ball was put back in play. At the next time out, the observer says to me, "You. . . you're pretty good. But you're
stupid!!!" Now, I could've said "But Edgar, I slid back into position, I fixed it, blah, blah, blah". But I didn't. I kept my mouth shut.
You learned an important lesson, albeit an unpleasant one. Remember it. Have a great season.
Chuck