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

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

There is no way of correcting a fellow soldier for disregard of army regulations than by being aggressive. You should also correct the Major by telling him or her that you are a professional soldier who hopes to become an effective NCO and that you are duty bound to correct all violations of established regulations in that endeavor. The Major was wrong in not correcting the PFC himself. You are to be commended for your actions in this matter.

I think you should have been more aggressive! What is up with the Pfc? He/she should have been doing push-ups!!!