What’s the proper way to correct someone when not saluting an officer?

You were in the right and the Major was wrong, That is part of our problem- Don't make waves, don't hurt someone's feelings.

I would say "hey PFC, can I talk to you?" Then step to the side and remind the PFC that proper customs and courtesies require the greeting of the day and a hand salute.

Maybe the soldier was having an off day or had something on her mind. Always try to assume the best, but don't ever be ashamed of holding your peers and subordinates to the standard.

You could explain to the MAJ that you were doing the PFC a favour. Just because he/she didn't care, it doesn't mean the PFC won't come across an officer who does and won't hesitate to dress her down!

When I am walking with an officer that does not get saluted, regardless of how the officer feels about salutes, I tell him I'll catch up in a sec, then turn around and address the issue with the Soldier quickly, quietly, professionally and out of earshot of the officer.
Praise in public, and more importantly, correct or admonish in private, regardless of rank.

The rest is just my personal philosophy, your mileage may vary;
Is saluting all that important on the spectrum of what we do day in and day out? Probably not. But it is probably the easiest one. And I don't really care what kind of day they are having, because if I can't trust a Soldier to do one of the easiest things on the list just because they might be preoccupied with something, how can I trust him to get anything at all done when we really need it and nothing is going right, you know the kind of situations that the military exists for...

If they though you were aggressive I’d hate to see what they would think of my approach.

SPC (Join to see) was the Major a direct commission? Sometimes they are either overly dismissive of military culture or all of a sudden insistent on it. From what you wrote, at face value, you did the right thing, on two levels. One you spotted an infraction and took some action. Two, the junior in the group is usually the best to make the on the spot correction.

I once saw (from across the street) the CG of NTC walking with the Commander of 11th ACR (Blackhorse 6 is an O6) and the Hospital Commander (an O6). Two PFCs from NTC Support Battalion walked right by all three. Blackhorse 6 had to jump in their shit. And you could hear it throughout the grid square.

Your delivery may have been perceived as bad. Perhaps pulling the guy to the side would be the absolute best COA, but no need to make the day of a peer to peer on the spot correction. I would have asked the Major what he would have preferred. Every time you walk by something ate up, you just set the new standard. Has nothing to do with that officer's vanity or degree of comfort with on-the-spot correction. That soldier should be glad it was you and not their CSM.

You were exactly correct. You spoke in exactly the same manner any NCO here would. If it was my children who did something similar I would have said the same exact thing. What you were doing was NCO business and you were doing it well.

I'd say your wording was needlessly sarcastic rather than aggressive. Obviously you know that enlisted still salute officers, so why ask the snarky question. All that does is create an immediate "what an asshole" perception in the person you are correcting. And IMO there is no need for that. It's great that you made an on the spot correction, but do it in a professional manner. Just stop the soldier and point out her error and tell her it's important to be aware of her surroundings and act professionally at all times. But saying things like you did, or an old standby ( yelling "hey troop, you got a broken arm?!") doesn't really project a professional approach either.

Thinking back about medical officers and saluting made me laugh. I had a next door neighbor at Ft Knox when I was there as a CPT for school, who was an eye doctor newly commissioned as a CPT. One day we were walking together in uniform and a soldier walked by without saluting. I immediately made an on the spot correction. When done, he said to me "I know I'm supposed to do that, but I never can get up the nerve to do it".

Praise in public, punish in private.
"Pardon me gentlemen, I need to take care of something real quick" excuse yourself
"Hey PFC, got a second?" pull him aside and make the correction out of earshot of the officers.
Aggressive is Subjective, however if the correction is done in private then that subjectivity is removed or even better it contextually isolated back down to enlisted who will appreciate the specific approach. Don't get me wrong, you might still be aggressive, but it will be out of sight allowing you to make the necessary investigation and adjustment.
After you are done, catch up "By your leave" render honors, and if asked "spotted a lapse in discipline that needed immediate fixing."

Good job, as long as you weren't screaming and making an ass of yourself. There's nothing wrong with being firm and assertive.
As much indiscipline as I'm seeing around and no NCO's making on the spot corrections, as a retiree I'd have probably phoned in a positive spot report to your 1SG for observing a SPC taking decisive action.
Thank you for making an effort to do the rite thing.