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

You were correct. The Major should have acted, sometime I would say " good morning private" or something to that effect to snap the soldier out of their "trance" and move my right arm to start the salute. This worked well most of the time as failure to salute is usually not deliberate. Some of the duty uniforms now do not emphasize rank insignia and creates a problem.

On this day 43 years ago I went through my first day at Recruit Training Great Lakes, IL.
In my time, You'd gotten a LOT more shit. At my first non-training command I worked in Security while waiting for a school on the NRMC Portsmouth campus.
The Security Officer LT/MSC was a Grunt Corpsman in Vietnam, and when walking around had no issues of pulling exposed gold chains and ear-rings off Corpsmen. Had a Jewelry box full of that stuff. I think if someone had passed HIM without a Salute, I think he would have just bitch slapped them into the ground.

The Major was wrong and you were not aggressive. I would like to know what the PFC gave as a reason for not rendering a sharp salute.

Immediate, direct, and to the point, off to the side if possible, with no yelling or drama. Keep emotion out of it. That said, this is grown-folks business, and hurt feelings don't matter.

Years ago I participated in a joint service exercise. I am USAF and the CG was Army. For my job I often wore civilian clothes. One day I passed the CG and did not salute as I was in civvies. An Army Major in his group pulled me aside (which I aprecciated) and then jumped all over me about it and after a rather colorful tirade ended with this thought which has never left me- He's a General in the US Army. After all that he has done to work up to that rank I should think that his service to his country alone deserves at least the respect shown by a simple salute. This Major was not overly concerned with my feelings, and he was quite aggressive. And he was 100% correct. Don't go soft on the young troops, teach them well and they will serve well.

from my personal experience, it depended on the officer's reaction. If the O started to blow his top, the junior in the group should pretty much snatch the enlisted aside and get him/her away from the officer in question. THEN you correct the attitude in whatever manner might be proper. If the Officer generally ignores it, I would say something like, "excuse me sir, i'll be back in a minute" then bend the offender's ear in a reasonably polite manner to find out what his/her problem is. Jumping on the offenders shit right there in an aggressive manner is almost NEVER the proper response.

nuts cover your ass at all time

I, was a Marine E5, one morning in formation the Captain, informed the troops, we did not e to salute all day long, a new order handed down from Hdqs, directed if we fall out for formation in the morning, and saluted the Commanding Officer it was a given we did not have to salute all day long, that order was sufficient, two days later a 2nd LT leaving the mess hall reprimanded me for not saluting? So if in doubt salute.

If they aren't one of YOUR troops, a simple "PFC, the custom dictates to salute officers" would have been in order (in a professional non aggressive tone). If she was one of your troops, same words but follow up with a verbal counseling in private at another time.

The counseling that I always used was a 5 minute explanation of the heraldry behind the origin of the military salute and an explanation that you are committing to a custom to pay homage to this ancient custom by saluting the rank. Then I had them sign a memo for record, which I kept in a small pocket notebook that everyone knew that i kept, and would never show up anyplace else.

I’ve saulted first. There’s confusion over where non salute zones end especially at that rank. I forgot to return a salute on commissioning day. That was embarrassing.