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

The major was wrong, twice over if he said it in front of the pfc. Unfortunately your response should be to suck it up in that instance, but not let it effect your own professionalism. And I'd try not to walk with that major if possible, unless of course your job requires it, just in case the situation ever arose again. Coming from the USO, if you were fraternizing this is one of the pitfalls. I always tried to keep a respectful distance even when walking to the same place.

as a private i walked everywhere i went, sometimes long distance to and from work. one day i was walking and i noticed someone in the distance but my mind was elsewhere, i was always thinking when i walked and sometimes prayed but that day i heard someone yelling PRIVATE, i stopped and looked around and there was an officer on the other side of the street looking at me. she asked me if i was not going to salute her which i did and thought to myself was this really necessary because i did not noticed you. but it taught me to always look when i see someone no matter how far away they were and if i could not see their rank clearly i would still salute, i have had some strange looks from people that was not officers but i do not like to be corrected and i never wanted to be corrected for the same thing twice.

As I learned at the NCO Academy, "You praise in public, you 'instruct' in private". In a situation like this I would have told the PFC to "stand fast", requested leave of the officers, then spoken to the PFC about "customs and courtesies".

Interesting; though, a question for SPC(verify to See): Does this Hospital[Med Cen] have a "No Salute Policy"? Yes, such policies exist, and the Hospital Command informs the personnel assigned of the policy as well as informing the Commanders of Units that share the installation that such a policy exist. Now; that 'sphere of influence' for the "No Salute Policy" is only in effect within the immediate vicinity of the Hospital grounds and if the USO was inside of the 'Sphere of influence' ergo...
So:
1. Does a "No Salute" policy exist in and around the Hospital?
2. Is the USO a part of the Hospital AO or to the Installation?
3. Was the PFC wrapped up in their "Dumb Phone" and not observant?
4. "Brownie Points"? Just Ask'en.
5. Yes I'm being a 'Smart Ass' but, Fitzsimmons AMC, 5th Gen Hosp, 130th Station Hosp, 2nd Gen Hosp.[Landstuhl, Germany] as well as other Military Medical facilities around the globe, have/had a "No Salute" policy.

You were correct. The Major was wrong.

The MAJOR and Navy LT should have quietly taken this to the PFC's chain of command and the company commander and other officers should have talked to that person in private. The first question is....do you understand the proper protocol when an officer approaches? If yes or no, the next question should be do you like being in the military? If yes, the PFC is sent to a class on proper protocol when approaching an officer. If no, start paperwork on the PFC....end of story....officers have the obligation in this case to address a young PFC who may or may not know or be unsure of proper protocol....calling the person out in public is going to embarrass and likely have a negative effect. The PFC during the meeting with the officers has the opportunity to make things right and grow in leadership or act like a child and be thrown out of the military...this is not the place for such a person....that person should start at a kid level....flipping burgers, learning customer service and build from there.....the job of the officer is not just to receive respect but to earn it further by helping people such as this young PFC. If those officers do not see and understand that, they should NOT be officers...they are leaders....not particularly upset when they don't get saluted or recognized properly. The salute is important to military readiness. I'm hoping that today's officers and you understand that the salute has a particular purpose....to show the person being saluted not only respect but that the person saluting is unarmed. I would like to see some officer input into this....my military career though not a full career was enhanced with officers and GOOD NCO's who were not "snowflakes" who got their feelings hurt easily but showed maturity and leadership when they detected something with me that needed addressing....the military is not about "good feelings" and everybody being on an equal standing....that is not the case in the military....ranks are there for a purpose....the military is about protecting the nation, about being battle ready, ready to organize and move on a moment's notice....let's get our collective heads out of our "politically correct asses" everyone....Obama's military needs to go....

My experience was that Medical and sometimes flight officers are a little lax about military ettiquette, or were in my day (Vietnam). I often saluted officers and didn't get a return. I don't think it was an aggressive response. I heard far worse of a response for failing to salute, like a total dress down. But it usually came from an NCO. But as an NCO, I was taught it was more productive to praise in public and criticize in private, which I always tried to do.

Just nicely remind them not to forget their customs and courtesies. It seems your initial response, while not necessarily wrong, came off passive aggressive.

How about: "PFC, you are required, not requested, to salute officers, why did you not?" After the answer, reinforce the requirement and then require the salute and return it. If there is guff put her on report. 4 years and 4 months in the Marines, NEVER saw this happen, NEVER saw disrespect, the only time I saw an open dress down was when a SgtMaj dressed down a Captain for improperly addressing him, called him the forbidden "Top." I read a lot of nonsense on here from other services....…… you guys are different animals, the Marine Corps means business, "A" game 24-7. Seems to me there has been a lot of liberal influence in others, especially the Army. Sad. (Don't give me the "Marine biased" BS, my dad was an Army LtCol and I was an Army PVT before I demanded and got a lateral transfer to the Marine Corps. The reason...……….. the obvious amount of simple bull shit and lack of discipline was too much, I wanted a serious functioning, no BS experience and got just that. Proud to have been a Marine to this day, proud of the standards, proud of the traditions, E-5 on discharge, RAH!)