Disrespected while correcting a Soldier, how would you handle this situation?

I do not know what sex you are, but I would have "grown a pair" and told the civilian (respectfully) to cease, this is a military matter and will be finished in a moment. Counsel the prisoner, his guard and tell the defense council/presiding officer/prosecuting officer as appropriate.

To me, there is a right and wrong way to handle this. If you are that loud that a another person can hear you and is bothered by the noise from a room with a shut door, and you have no other place, then you might want to conduct this "after hours" so it will not interrupt others and gives some "privacy" (SP?) as well. If that cannot happen, then you might want to find another place in the building or outside away from common areas where you can have some privacy.

Call his chain of command and send them a Sworn Statement, with the witness' as well.
As for the civilian, do the same and file a compaint with HR.

Seems most are focusing on civilian and not SSG's first issue. Solider is at their court-martial in wrong uniform, when asked by SSG solider just turned away. Solider doesn't care about what SSG is stating and being in wrong uniform is just more "I want out now" what more can you do to me. SSG should make one of 2 choices, 1. With time permitting, march their a$$ back to living quarters and make them change 2. If time doesn't allow that when asked buy judge, solider can explain his uniform or more than likely do same to judge as to SSG. Don't needed to have an animism as your
health isn't worth a solider than will be out of your control soon enough. And I am sure the SSG has seen a few solider in the courtroom like that in the past.
As for civilian, As a GS-13 myself I would have entered the room and ask SSG to move to a more private setting to "correct" the solider. Don't need someone yelling in a public area as that is unprofessional. As a SSG you should know that and act like a NCO. There are was to handle people like that and bending to their level isn't one. And for the "Drill Sargent", See how fast your CO will be calling you to their office if you use that tone with me. Respect is a two way street....

I see a lot of responses to what they would do, personally I need more information on the situation.
Was he under your direct command at the time? Was he your troop? Did you just see him and decide to make the correction? What were both roles of this encounter? Who was up for the Court Martial? Let's take a logical look at this whole situation. We got the end of the story but not really the beginning.

You were right and if this soldier had an ounce of pride in himself he would have answered with respect and told you he was returning to his quarters to change. Maybe he didn't get the memo? Turning away from you was a definite sign of disrespect and he should be counseled for that too.

Just put it on paper...and place it on the growing stack of papers he is using to ruin his career with....

Reported the civilian up the chain of command and just told the soldier fine, BCD it is...

I think taking the time to cool your self down was a smart move. I know that when I used to be a corrections officer there was a time and place to unload on someone that wasn't following orders. So that may have not been the time nor the place. That being said though the soldier in question seems to be doing everything he can to just get out of the military on the worst terms possible. It's actually sad to see someone who probably at one point in time was so excited to take that oath and now to see them at this point. Makes me curious what causes that to happen in someone. I've always lived by the motto, when you feel like you want to quit remember why you started.

There's a fine line between not knowing and complete, utter non-compliance. Given the situation, I guess SM in question just didn't care at that point. Understandable, but to disrespect an field grade officer and senior NCO by blatantly ignoring simple orders, not to mention at a legal proceeding. Guess guy validated why he was there in the first place and future continuance in the US Army won't be much longer. I'm sorry Sarge but you and I both know people will be those individuals. And added, maybe the location was a little out of pace since you might have conflicted with others ability to do their jobs, but that's not a knock on you. At that time, place and juncture, emotionally you felt slighted and you needed to do a on-the-spot correction. So you did what you felt you needed to do at that exact time. Soldier just took it to another level.