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

There is no other way but directly and to the point. Since when has the soft touch become part of our military. You speak to warriors with strength and civilians with kid gloves. I'm surprised you didn't drop this private. You're not gonna do our country any favors by treating military personnel like civilians.

More than 50 years ago I was an E-5 patient at the "Old" Walter Reed for followup treatment after eye surgery for the wound that got me a Purple Heart in the Nam. Of necessity, both eyes had dilating drops daily. One sunny afternoon, when I was walking between buildings, wearing clip-on sunglasses over my regular sunglasses, and with tears rolling down my cheeks, I was stopped by a Medical Corps Captain (female, not that it matters) and given a royal a** chewing for not saluting. Circumstances were, I thought, pretty obvious and that was one case where the officer might have let it pass.

I would have asked politely why she didn’t salute, and then requested the salute. If she still refused to salute then I would have requested her CO’s name and rank and talked to him about it.

As an Enlisted member I can say you were in the right. Sadly, there's no way of correcting someone without being a little bit rude.

In Iraq, a soldier and I were walking into Al-Faw Palace when General Cone(3star) at the time and his entourage were walking out. I saluted this wall of blinding brass but he didn't. I quickly corrected him on the spot as well as had him run past and salute. We did our business in the palace and returned to our office. For further correction I had the soldier stand outside our HQ and salute and give the greeting of the day to every NCO and officer for about an hour or so. During that hour a Maj came in and ask why this PFC was out there doing that. I explained what happened and he said I was too excessive. I politely reply "Well Sir, there is Officer business and NCO business, unfortunately this is NCO business" He promptly walked out of my office and watch my soldier salute and give a greeting of the day.

Long story I know but I believe your intentions were correct and with time you'll know how to handle these situations differently in the Unit you belong too.

I can remember not being saluted. The E3 (Air Force) boss a Major decided that the E3 “forgot” he had to salute officers. Both we’re African Americans. I just wrote it off as an Anti White respect thing. From that point on I had no dealings with the Major.

Since you stated "My hospital has both services", you may want to check the "No Salute" policies in the area. Some military hospitals have no salute areas because of the large number of officer and enlisted. Just because no salute is rendered, it is not necessarily disrespect.

Yours was a proper question. The Major was incorrect.

I to once had an incident concerning not saluting an officer. I was walking to the chow hall against the wind, and was leaning forwards into it and looking down. A major was walking the opposite direction towards me and stopped me as I was passing him. I lifted my head up as he said "excuse me, but you forgot to salute me", as he pointed out his ensign on his collar. I thought that was tactful. I apologized, and saluted him. I walked away appreciating the fact that he was polite about it. I've heard stories that some officers can be pricks about it, and others can careless. What can you do, lose sleep over it?

I was an 2LT walking on the other side of a 4 lane street at Shepard. I didn't recognize a Captain walking on the other side of the street until he called out and walked across the BUSY street , dodging traffic, and spoke to me "loudly and sharply" (I don't "yell" at people).
I cane to attention, saluted, and said that I had NOT recognized he was a captain. He said , since I was a 2LT, that I must "be new", "(I had a number of service ribbons. 3 rows probably.")