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

If they are in the military they should clearly know the rules and you did he right thing to teach the LT and Maj plus her the rules which she should have known. As a CWO USN, I stopped other officers who didn't have their jackets zipped up according to rules; I was very respectful and stated, If you don't set example then how can you say "you lead by example" or tell your troops to obey the rules. I stopped p to 05/06. Guess what, they looked at me and was surprised but ended up saying thank you and zipped up their jacket to the required level. When you are right, you can't be challenged. But you must be respectful and not embarrass them. I suspect they never forgot that moment. PS. I also always set the example myself.

Well done, Specialist. You saw an infraction and enforced the standard. Carry on.

What I do know is that in the Navy any correction would have started with, "Hey, Shipmate......"

I think you were fine. The Maj in my eyes was wrong in a way. That PFC didn't adhere to the regulations and you as a higher ranking individual corrected that. As an E4 at that most E4’s congratulate that PFC. Good on you!

I would have addressed the pfc with tact that officer understand. But as well I would have the pfc go back and give the proper respect the pfc is trained to do in basic and by regulations

Good question, and you show humility in asking.

Here's the answer from my perspective. You were right on the money. You're clearly, already conducting yourself as an NCO. When there’s an on-the-spot correction that needs to be made, that should be made, and that can be made, light them up. NCO’s A R E the military. The rest of us are just here to fall in on your example.

Coming off aggressive? I don't know about the medical corps but as a former infantry platoon sergeant I'd say that was just about the right respnse

You were definitely correct to address that young man. I have noticed that the Army in general seems a little bit more lax on these courtesies and customs than what I recall in the Marine Corps. I would have pulled the guy aside and reiterated (he should already be familiar) courtesies and customs and explain to him that the next officer may not be as lenient and if a high ranking officer he's unfamiliar with happens to be in his chain of command, it could have ramifications. He needs to remain cognizant of his surroundings particularly while in uniform. Just remember that as a leader it important to practice tact; you want to correct soldiers, not necessarily embarrass them in a harsh way. Depending upon the circumstance, this can be difficult to avoid but the goal is to make junior soldiers better.

This should be addressed aggressively. ",Soldier: stand at attention. You have passed an officer and have failed to salute. Execute a proper salute on my command: execute. Do you understand the importance of this? Outstanding. You are dismissed."

I can't imagine a private would be so careless. Especially since it hasn't been all that long since they were in basic. Nor can I recall ever missing a salute. It just wasn't done. But then I'm old school. As for tone of correction, if it's same sex personnel then out of ear shot is fine if it's opposite I'd more likely remain within sight and hearing of any group so there are zero misunderstandings. No need to be all hostile when correcting any rank of a glaring infraction. Sometimes its "walk quietly carry a big stick"! It can be a carrot or an actual stick
(EMI) actions as warranted by attitude of one committing infraction.
Easy.