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

I think that sometimes, situation depending, making someone feel a little guilty or embarrassed is better than an ass chewing because they don’t want to get embarrassed or make the same mistake again.

“I’m sorry private, I must be jacked up if I don’t rate a salute. Don’t worry I’ll fix myself. As you were.”

I think you handled it fine. Though the Major should have said excuse me PFC didn't you see the LT and I?

You did the right thing. As sometimes happens (I'm assuming the Major was medical services) they're not used to the basic military customs and courtesies of any service since they were probably direct commissioned and later had some quick shake and bake school. If the Major was not direct commissioned then it was a flat out mistake on his/her part to be that direct and, to that point, you just say "yes Sir/Ma'am" and carry on bringing the minor exchange to your CoC.

The last time I checked the regs., all enlisted, up to E9 will salute all officers, yes including 2LT. However, as our society and military moves towards a more "sensitive" military, one in which we don't hurt anyone's feelings. We forget to enforce basic military courtesy and respect, for fear of a potential IG complaint. Call me old fashioned, but I learned a long time ago while serving in the Marines, that we salute the rank, not the man/woman wearing. And yes, the major in the posting was/is wrong; it is that precise attitude that landed us on this posting to start with...

Saluting is an exchange (greeting) between members of an honorable profession. Choosing not to salute or not to return a salute is just dishonorable and disrespectful. When I was a Tech Sgt (USAF), I encountered just to opposite where an enlisted member saluted and two medical officers turned their heads. I very "politely" approached the officers, with salute I held until they returned it. Then I prefaced everything I was about to share with the magic words, "With all due respect gentlemen." The rest of my talk was not very polite. Then I saluted, turned, and left with their due respect.

You did exactly right. The Major was in the wrong. It's unfortunately part of this newer "feels first" military.

Aggressive would have been walking back, locking the PFC into position of attention demanding name and unit and then turning it over to their CoC to correct and re-train.

The only thing I would have done differently was I would have taken the PFC off to the side to do the corrective action. Other than that I think you were perfectly right in what you did.

I actually had to handle a situation or two like this. The way I addressed the situation was, "Hey PFC get back here". When the PFC returned and saw the officer he rendered the proper salute. If he hadn't, I would of required that he do so at that moment. Then I asked "Why the hell didn't you salute the commissioned officer the first time when he was so damn close he could have bitten you? Now IF, by chance the knucklehead works in an environment like the Pentagon where half the services don't even wear a cover and there is no saluting, you may understand their ignorant behavior, but then remind them that they are not there anymore and they need to pay more @#% attention to their %#$@% surroundings! ("Of course I may have used more colorful language back in the day so as to not make the situation boring.)

Continue saluting. You are showing respect to the rank and not the person

I would tell them be more aware of their surrounding. It doesn't matter what branch a strip is strip, a Bar or bars, oak leaf, FUll bird and a star is a star in any branch. When In doubt salute and away