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

We have a number of issues here. Chances are you were with individuals that are in the medical corps. We line officers refered to them as 90 day wonders. They spen 3 months learning drill and ceremonies and this was predicated on their background in medicine (You can also insert JAG, area defense council & chapelin here). This is not an excuse, just a reality based on these individuals not being combat officers. The whole movie "MASH" and the television series of the 1970s revolves around the idiosycracies of military medicine. That said, yes, a saluet should have been offered or reterned. One of the best ways of handeling this would be to consult with your first sergent and have him mention it to the medical corp first sergent, where it will then be elevated to the hospital commander who will send out a memo to all medical personnel to observe military cortosies.

For most marines in my experience it was something like:
"HEY THERE DEVIL! I GUESS WE DON'T GIVE THE PROPER GREETING OF THE DAY NOW!?"
Much respect to all of you for your restraint and courtesy.

You did good! I would’ve lit them up, it’s the military not McDonald’s

In 1970 I was at a rear-echelon Army base in Vietnam. I was a Spec-4 (E-4). I arrived at (believe it!) a Tennis court to play tennis against a Captain in my unit. We worked alongside each other. I forgot to salute him. Without hesitation or delay he matter-of-factly reminded me to salute when approaching or passing an officer -- especially away from the "workplace". I instantly did so and learned a valuable lesson. Don't get casual. Maintain your decorum no matter what, as it's clearly important to the maintaining of military discipline at all times whether in uniform or not. Undisciplined behavior while away from the combat zone, like weak training and individual/unit discipline, erodes our readiness and must be avoided "at all costs", as they say.

I think you're completely in the right battle. Although the MAJ may not have liked your style as a future NCO its your job to maintain standards and discipline. I don't see what you did as toxic leadership or a bad way of addressing it. Good work.

I don't think you did anything wrong. You can ask many different ways but the bottom line is, the PFC was definitely in the wrong. I would've handled it slightly different by asking the PFC to chat off to the side. I would've asked the question about saluting and if he/she knew what the regulation was and if they could explain it to you. It also depends on whether they were looking at you all and just blatantly walking by - that would definitely change my approach.

Hate to say it but nurses and doctors generally do not care much about customs.

Keep your head down and move on. If you are right but the seniors are wrong, very little can be gained on trivialities. If the stakes are high and the results matter, that's another story. But, at the end of the day who does or who does not salute whom really doesn't matter in the bigger scheme of things.

Your "question" was hardly aggressive, I would say it was a mild response. As a retired Medical Service Corps officer and former Corpsman, I know that medical commands are often lax re saluting.
I also know of several medical department officers who are not "into" being saluted.
The underlying problem lies in the poor military training given to our medical personnel.

I walked by a Marine Captain and an Air Force Lieutenant when checking in to my "A" school fresh out of boot camp. The Captain asked me if I saluted officers, and I said yes and did so. He and I later became friends! I had all the Naval Insignia down pat, but this was a combined command. At the time every Airman or PFC looked like an officer in their dress uniforms to me, but I quickly set about learning what officers from the other services looked like!