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

As has been commented, How you correct someone is most often more effective. I am a civilian now and have had many opportunities to correct people in how they conduct themselves. I have noted that how I correct them makes a huge difference. For something such as saluting an Officer being agressive is not necesary, a simple reminder that in acordance to military customs an enlisted person is required to salute. It is a sign of respect not necessarily of the person but of the rank. As a civilian a handshake and or greeting is just as important. Look the person in the eyes and show them the same respect as you would want to recieve were you in their position.
The PFC should have been paying attention to what was going on around them and rendered a salute as per regulation. As junior enlisted to yourself, however you corrected them should not have been any business of the Officer. Attention to their surroundings as a military member is and should be a high prioirity so when they find themselves in a combat situation they already have that lesson learned. It could keep them alive.

If the US ARMY wanted you to have feelings, they would have issued them, correct the problem, loud and proud, so they will remember it and it will not happen again

How I would have handled it.

To Major: excuse me Sir, I'll be back in a minute.

:Walk to PFC:

To PFC: Hey PFC [name]. This isn't me jumping down your throat but this is what I noticed. (Proceed to explain)

In most situations I try to give everyone the benefit of the doubt.

Leadership begins with leading. If the PFC failed to salute a leader will correct the situation, calmly, with clear purpose and without animus. If the PFC felt emotionally stressed by a reminder of standards the PFC in question will benefit from an improvement in maturity through experiencing stress and surviving it. As I recall from my days in the Navy the usual standard in such situations is that the Junior Officer or NCO present performs the admonition which permits the Senior Officer to ignore the slight. As to the "aggression" - please - we are in the military - aggression is the very center piece of our profession, our soldiers should be able to handle the stress of a mild rebuke when they screw up.

An on the spot correction is necessary and I always used the tone of voice that let the guilty party know if best never happen again in my sight.

This is a very hard question to answer. Trying to correct the PFC's misconduct in front of others and in public. Should the be called out in front of the officers or taken aside and and corrected. In all my years of supervising others I can't think of a time I corrected an officer or staff member in front of others unless it was in dangerous or life threatening situation. I was a supervisor in in a prison for over 20 years which had military type ranks.

This situation calls for good old fashioned wall to wall counciling.

Would also remind the LT and Maj(respectfully, of course!) that saluting isn't about courtesy, it's about respect and not getting a salute is disrespectful. If you don't respect yourself, neither will anyone else. I notice most medical types tend to be less stringent on those kind of things.

There was nothing wrong with asking that question to the PFC. The PFC could have gotten destroyed by a Sr NCO. Now with the Major you need to know which battles to fight. Officers will speak their mind and you have to listen, however you were not in the wrong for attempting to correct the PFC. So keep upholding the standard and pass that on to the soldiers in your section.

I can see that it is a much kinder, gentler Army then when I was a young officer in the early 1980s. This situation has happened to me many times. I always tried to be professional, yet direct and most of the time the soldier took corrective action Without incident. On occasion it was necessary to jump their ass and in one instance I even took three young soldiers to their first sergeant who was a friend of mine and let him “re-educate.”