Are you allowed to walk away from leadership when they being to act unprofessional?

In short, no. Disrespecting an Officer or NCO is really left up to the judgement of that Officer or NCO, so walking away could be viewed as you disrespecting them or outright disregarding them. Neither of which end well for you. However, after the fact or if it becomes too much, you can always exercise the open door policy of the person one step higher than them. Best to have a battle buddy that can confirm whatever it is you think is unprofessional that they said or did, ie cursing in your face, calling your mother a whore, whatever it is.

Best bet is to remain professional and calm, roger sir/sergeant, then take your battle buddy and go see immediate chain of command. Do not exaggerate and do not lie.

Depends on how much you like extra duty.

True story: Back when I was a PV2, my squad leader was a complete ass. After getting smoked for the umpteenth time for no reason other than that he saw me (literally he would drop me just for being seen), I recovered myself and walked away. He started hooting and hollering, and I just walked to my barracks room and locked my door. 15 minutes later, 1SG was pounding on my door. Here is, paraphrased, what top told me. "What you did absolutely needed to be done. Unfortunately you don't have the rank to back it up. Tomorrow you will be seeing the Commander for your Article 15." We proceeded to have a gentlemanly conversation about the squad leader, the merits of a court martial, what to expect in an article 15, and how to move on in the Army. In essence, 1SG had pity on me and my situation. I was right in theory, but because I was a PV2 and he was a SSG, I was still wrong.

And that's how I got my FIRST Article 15....

Allowed to walk away, No. the advice from SSG Marc W. and SSG Carlos Madden is spot on.
One point to add: NCO's are people too and most, not all, but most require some infraction before their ire is up enough to cross the professionalism line. That being said, ask yourself this one question, what was the cause of this NCO crossing that line? Could it be that there was something you did that could warrant a reprimand? Perhaps not in the ranting/raving way this NCO presented, but check yourself first...then consider what the two fine NCO's mentioned above relayed to you.