How do you handle a disrespectful junior soldier that doesn't belong to you?

There is a time and place for reason and discussion, and there is a time and place for other forms of communication.

If I was in your shoes. I would have called the MPs and had him arrested for disrespect and then they would have to contact his unit. I would then go to his unit and report him to his first Sargent.
I am pretty sure Top wouldn't want his unit representation like that.

From your explanation, I take it this took place off base somewhere and that you were in civys. Unless you can identify the man, find his unit, or find a cop willing to step in, there's not much you can do. With so many people in the communities wearing uniforms, the stolen valor actions happening all over the place, and the mental issues that people are claiming and actually suffering, there's not much you can do.

If you're on post, easy.. Call MPs and just keep him in sight. Off post, where appearance in public matters, … yeah, video and get a tag and follow up with MPs or Base CDR/CSM. Thirty years ago?…. tag him till he's shaking on the ground.. THEN call MPs to pick him up. He's probably POS civilian stolen valor… which in some places is a misdemeanor ! Citizens Arrest!

There was a day when Wall to Wall counseling with dimension lumber would have cured the issue, but I'm afraid we now have a kinder, gentler Army.

I think you did the right thing. You went through the steps as prescribed. Two incidents happened which shaped my thoughts while I was in.
The first, I was a company commander and we were running an APFT when an officer passed one of my road guards and my soldier made a comment. This officer noted the chain of command, and I went and apologized face to face for my soldier, then brought her and her leadership in for a formal counseling session. Humbling for me, but I did what was right.
The second incident occurred when I was the S3 for a training BN at FT Lee. My CSM was returning from a meeting at the post HQ and noted a couple of soldiers who were not to standard. He stopped and made spot corrections; note, he did not yell, swear or in any other manner abuse these poor troops. In the five minutes it took for him to return to his office, he already had a message from the garrison CSM because my CSM didn't go through the proper chain of command to correct the troops. I'm still in disbelief that a CSM was called on the carpet for making spot corrections.
I'm not really sure what's happened to the Army in the last 20 years, but there's those who should be there, and those who will not last... I hope.

Take off your blouse and go to the tree line. Or go to the Px buy the same rank, go back and find his azz.

Had a similar incident in Nurenberg Germany back in the early 70's. I was a Staff Sergeant at that time and the young private was telling me things about my ancestors I didn't even know. I would have like to light him up but I followed him to his unit and reported his conduct to his First Sergeant, after which I was summoned to make an official report to his commanding officer and he was given an article 15. You violate every regulation in the book if you decided to go fist to fist with this troop, but the best thing is to follow him to his unit and if that is not possible, get as much information on him as you can. In some instances you may just have to grin and bear it, but things like this happen. You never put your career on the line to straighten out a troop that is not yours. If he is disrespectful to you chances are he is the same with members of his own unit. Troops like that think they may be getting over being disrespectful, but eventually he will hang himself at his own unit.

Well, I recommend a quick kick to the nuts, followed by calling MPs. They can figure out whose unit he belongs to.

I get the vibe this "soldier" was anything but. There's been a rash of incidents where some guys wear the uniform to get discounted meals, try to score with women, etc. Call him out on it and see how he reacts. If he CAN prove it, then you know what to do from there. If he can't, he's a faker and all you need to do is just let his identity be known throughout the area and let karma worker her justice.