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

You were right and officer was wrong. I would have corrected the EM, sent him on his way and the I would have talked to the Major and informed him "with all due respect, Sir" the next time I am correcting someone please do not interrupt.

The Army Maj was wrong - there is nothing aggressive about requiring the customs and courtesies be enforced. The PFC wasn't paying attention, given the benefit of the doubt, not out of disrespect so it was just a wake up for her.

A soldier should always (quietly) pull the soldier aside and remind the soldier of proper military customs and courtesies.

The major was wrong! Hospital units are by nature relaxed in saluting. A lower rank should always salute an officer of a higher rank not only courtesy but historically pride and institution.

'Aggressive'? The Major though you, and that, was aggressive? I'd have been, "WHERE IS YOUR SALUTE, SOLIDER?"

While serving in Korea in the early 80"s on a joint assignment I was walking with our Sgt Maj and an Army troop failed to salute me USAF Officer). Before I could say anything He said let show you how it is done. He proceeded to in a very calm voice to dress the soldier down and directed her to apologize and render the proper courtesy. Since it had become a problem with Army junior enlisted not to salute other branches and countries officers the senior NCO were marking it a point that infractions include a very public on the spot correction. After the incident was done the Sgt Maj turned to me and said the next one was mine. It did not take but another 100 ft and I got to demonstrate what I learned from his example. As he put it there is never an excuse to be unaware of your surroundings.

Well i don not know about the other branches, however, in the much RESPECTED MARINE CORPS that would have been an immediate ass chewing with the utmost intensity.

First ask is your arm injured? When the reply is no the rip a new one. It reinforces attention to details and situational awareness. Happen to me as a brand new 2LT and I never missed a salute for the next 25 years!

This is b.s. and a part of the lack of discipline we are seeing in all branches today.

I see nothing wrong with the way you handled the situation. Too many courtesies and customs are ignored or disregarded without consequences. There’s nothing aggressive about holding people responsible for their actions (or inactions).