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

I have personally heard stories from a World War II Veteran about this same issue. He failed to salute an officer so the officer gave him a spech about how it is not the person in the Uniform but the Uniform you will salute. Then you put a officer's uniform on a tree and had him salute that uniform on the tree all day.

On the spot correction. Who needs to ask? You handled it! Aggressive is "on the spot correction."

it depends on the AO SOPs check with your first line leader for details ...

You got the point across that’s all that matters.

You didn't do anything wrong by what you stated. You made an on-the-spot correction, which the Major shouldn't have corrected you on, but sometimes you have to roll with it because every officer reacts to such situations differently.

Dont sweat the Maj, he was more than likely a direct commission medical officer. You did good. I like to start my corrections with a "Hey you!" It gets there attention and they usually realize they messed up before I start talking about it.

Nothing wrong with your approach at all. Not saluting is a sign of disrespect and a violation of a regulation. Had you waked by and not corrected the Soldier you would have become part of the problem. Thank you for your service.

Most of it has been stated. There’s a way to be more tactfully aggressive. However, a good warning/piece of advice moving forward is that, if you ask Privates ridiculous questions, you have to be prepared for ridiculous answers.

By that, I mean, the idea of your question wasn’t ridiculous. Of course, enlisted still salute officers. However, what if she pulled the dumb Private card out and pretended to ‘me no speak Army’, and answered along the lines of, “well actually Specialist, no. No we don’t salute anymore.” (If you don’t think that’s possible, go ahead, keep asking random Privates ‘yes/no’ questions). Your head would have almost exploded, some sort of argument would have drawn out, and if hypothetical Private Knucklehead wanted to do so, she could have driven you crazy for at least a few minutes. Your officer pals would either have to leave you behind, or wait for you, either making their jobs a little more difficult when they got to where they were going without their enlisted bulldog, or simply wasting a bit of their time.

Of course, the above scenario is a bit hyperbolic, but go ahead, keep asking Privates crazy questions. You’re going to start getting crazy answers.
(If I were you, I’d probably have excused myself, addressed the Private, and caught back up).
Best of luck.

I've been in a situation where I made that error. I was a sgt. My private was being a problem and ran full speed away from the company area where we were working. In the process of chasing the private I did not identify two full bird colonels that I ran within 6 steps saluting did not happen. I also wasn't questioned about it.