Conduct unbecoming, if nothing else. It may not be a prison worthy offense given the cultural extenuating circumstances, but it's clear that regardless of the internal policies of some UFP member worlds, Starfleet itself does not condone dueling to the death, as evidenced by Spock's expectation to be penalized for the events of Amok Time. That didn't happen, but much of that seems to be a result of the leniency & understanding of the commanding officer. Picard himself even finds a way around the death duel in Code of HonorWhile he did kill Duras, it wasn't murder under the Klingon legal code. What would you of had Worf charged with?
Similarly, I'd suspect that Picard is sympathetic to Worf's cause, because let's face it, Duras got nothing short of everything he deserved, but moreover, I suspect Picard specifically is sympathetic to Klingon values, on the whole, choosing to act on his behalf in a matter of his father's house , & even surprisingly falling in on the supportive side of his ritualistic suicide in Ethics.
So, by my reckoning, Worf gets over because of who his commanding officer is, & even then, he makes it clear that no such similar actions are acceptable, and by Worf acquiescing with his admission of not wanting to resign, he reaffirms that doing so is not an acceptable action by Starfleet officers
Some other captain might have thrown the book at him, with a court martial or even drumming him out of the service altogether imho