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

You were definitely exercising general military authority by correcting this Soldier. The MAJ is entitled to their opinion, but every Soldier is charged with enforcing Army regulation. The only suggestion that I would make is that Soldiers should be corrected out of earshot of others, and approached in a manner so as to maintain the Soldier’s dignity, and it should be an instructive moment such as, “Private do you understand that AR 600-25 obligates you to render a salute to all commissioned and warrant officers?” Conversely, an example of an appropriate time to correct someone within earshot of others would be witnessing a safety violation where someone may be injured or worse or when a group must be corrected for the same infraction.

Seems pretty appropriate to me, I do it with my LTs as well when it rears its ugly head. It all comes down to customs and traditions. None of us is above it and none of us is exempt. I've seen a CSM go about elbow deep in a Trooper. He certainly got his point across and the decorum between Enlisted and Officer was clearly re-established. NCOs and in your case a SPC hold the line. It's what we do and sometimes your point has to get across clearly. I respectfully believe you did the right thing.

It's not really surprising honestly. There are more and more millennial attitudes in officers nowadays. The officers in my unit do not like being saluted. But as for correction, I personally have a verbal than written mindset. Pull the soldier aside first and explain with a firm yet understanding tone about there deficiencies.

Is to call him and tell the person how important its to salute a superior officer and tell him not saluting could lead to charge

You did right. The Major should and the Navy LT should have made the correction.

I won't mention the General's name but our FA unit was at Graf doing live fire. I think this was around 1984 or 85. Our BC went to main post and while he was inside HQ the new BG who had just gotten his first star came walking by. The driver who was a private and a dirt bag did not get out of his jeep or render a salute. Result: that BG came to our firing point and for one hour he observed our battery doing D&C on proper salutes. Present arms......order arms, for an hour.

I once made the mistake of saluting a LtCol and Captain with a cigarette in my right hand. The Captain grabbed me by my lapels and slammed me against the wall while he chewed me out. I was a PFC at the time and never made that mistake again!

Senior Enlisted are responsible for training military customs and courtesy

Depends on the command. Many are top heavy with officers and don't salute....Hospitals are normally such commands... everyone is also called by their first names.... If the officers were not concerned ...you shouldn't be. Now with an infantry, unit that's a different story.

Passive aggressive. Turn, say Solider to attract attention, offer a hand salute. Accept appology graciously