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

You were correct to calmly address the PFC.

Praise in public, correct in private.

I had that happen a few times over the course of 30 yrs. I usually stopped them, rendered a salute and held it until they returned it. I would say "a salute is rendering MUTUAL respect for OUR service." Then I'd drop it and thank them for being in our Navy or Marine Corps. Usually it was a very junior person and they just didn't understand that we rendered courtesies like that.
Usually I got a "uh, sorry sir, I wasn't paying attention" or some other excuse.

The WORST case I ever saw was a Midshipman 2/c at USNA. He was talking on the phone, saw me and turned away from me (06 in uniform). You could see the anger in his face when I waited until he ended his call and held my salute. In his case I told him "We always render military courtesies unless in combat. Some day you'll earn an enlisted man's salute. Always return it crisply and with good humor, no matter how lousy you feel. He's earned your respect and if you shirk something like a salute you'll lose his respect."

I find nothing wrong with following military custom and discipline. CPT Wayne Price

It happens. Maybe the private wasn't paying attention or they had their mind on something else. I usually say, "Officer present." They look around, realize one is present (the look in their eyes says all), and salute on the spot. Gotta remember not all enlisted are disrespectful little ass-caps. They're human too and sometimes their brains are just hard-wired to something else. Maybe something's going on with their child, or they're thinking about school, or maybe it's work-related. You don't know (and it's really none of your business, right?). Just give a quick heads-up. Simple and no harm done.

Only time you should talk like that is when they're obviously giving guff. THEN they need to be put into place. Until then: innocent until proven guilty. Not the other way around.

The Major deserves to be corrected for his ill advice. The Salute is about mutual respect. I was an enlisted soldier, an NCO and an Officer, I saluted first most of the time. The Major is wrong

You do not salute in a hospital area. But being outside they should have but force of habit of not doing it inside probably took over.

That is the appropriate way to handle the situation

That’s the appropriate way to handle it.

Maybe it's my enlisted background, but as an officer whenever I fail to get a salute I salute first - this never fails to make the appropriate correction in the most mutually respectful and expeditious manner.