After reading both your post and your follow-up, here are a few things that pop out for me.
First, I believe that your son's apology needs to be an in-person apology. He chose to be rude and humiliate his teacher in front of the class. I've been in the military,and believe me, a note of apology would simply not fly. I've been called on the carpet by my CPO for doing something stupid-- it was an informative experience.
If he feels he can use adult language with his teacher, he'd best be prepared to deal with the consequences as an adult. A note saying 'sorry' for insubordination to an employer or authority figure won't cut it. The lesson will be hard, but likely, if you hold his feet to the fire, he won't do it again. I'm not sure why you are choosing the 'saving face' option when this could be a valuable lesson in humility and learning to clean up the mess one makes. Saving face, to me, would be scheduling a private time for him to apologize to her in person, not by allowing him to just write an apology without giving *her* the chance to respond, which I think is unfair to her and unrealistic for him. He spoke out and has to deal with the consequences. (Think about it, this would be like having two children who are playing, one hits other then says "sorry" without allowing the second child a chance to say "hey, don't hit me" or "I didn't like that".) Are you sending the message to the teacher that she just needs to suck it up and accept his apology without having her say? Is that a realistic lesson for him, going forward?
Part of making amends is having to listen, however hard it is, to what the offended party has to say. And given that her request to sit down was perfectly reasonable, I think she has the right to find resolution with him on this.
Second, you mention that you offered for him to talk to a therapist as an option. If he's doing what you are saying (circular arguments that go nowhere, rudeness, refusing to go to his room when you ask him), then he loses the *choice* of going to see someone. This is where we have to do the hard work of being parents and make the choice that he *will* be going to see a therapist. "You seem unhappy and we aren't getting on well when you are angry. I think you will benefit from someone else helping you deal with your anger so that you feel you have other options which won't have a negative affect on your future."
The whole "earning my respect" suggests he also needs a little help figuring out his place in the world. He may not like his teachers, however, they are not his peers. They are the authority figure in the classroom and there is a way to go about broaching problems he has with them. A therapist could also help him figure out strategies to problem-solve around this issue, so that when he does have a problem with an authority figure, he can think first and then approach calmly to find out how to fix the problem.
Of course, do listen to his anger and 'why'... and then, as others have suggested, then-- it's time to fix the problem, and then move on. Hopefully, forward. And I'd skip the grounding, personally. Unless he refuses to apologize to her. Otherwise, once it is resolved between them, let it be.
PS--If I seem a bit hardcore, please know that I've had my 5 year old son apologize in person to his kindergarten teacher (before class--we scheduled it) for a mistake he made that affected others. The apology and making amends was the consequence. However, the problem was between him and her, not he and I, so the apology was necessary. We have had no repeat incidents of that sort!