Why wasn't Kaffee charged?

Court martials are not some sort of magically fairy effect that shimmers into existence as soon as a soldier does something naughty. A court martial has a convening authority. Much like a civil trial, somebody has to file charges. After such a high-profile, emotionally charged case involving conspiracy and murder, it's unlikely that anybody is going to file charges for potty mouth.

You generally don't seen it in Hollywood courtroom drama, but there are often ancillary charges that go along with the "big" criminal charges. Somebody who commits murder is not only charged for the murder, but they'd also face weapons charges for whatever they used to do it, reckless endangerment charges if anybody else was around and could potentially have been injured, drug charges if they were on something while they did it, and so forth. Unless it was a habitual offense, a charge for language would be added to something more severe, not prosecuted by itself.

Conversation was like this:

Col. Jessep: ... I'm gonna rip the eyes out of your head and puke into your dead skull, you messed with the wrong marine!

Capt. Ross: Colonel Jessep, do you understand these rights as I have just read them to you?

[Contemptuously]

Col. Jessep: You friggin' people. You have no idea how to defend the nation. All you did was weaken a country today, Kaffee. That's all you did. You put people's lives in danger. Sweet dreams, son,

Kaffee: Don't call me son. I'm a lawyer, and an officer in the United States Navy, and you're under arrest, you son of a bitch. IMDb

As you can note Kaffee was not the one who started the provoking language neither did he seems intentionally doing it. It seems more like a response to the opposition's comment. So in that case, I think he can get clean chit or just a warning. Court-martial doesn't looks suitable here.

Note: It's a film and to get dramatic effect sometimes abusive words are used ignoring few facts, like mentioned by you.