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

Let me guess, the officers were medical types. While I respect their skill and their dedication to their job of taking care of all service personnel they have to learn to respect the customs and traditions of the military.
If you were senior to the private who forgot their responsibility to salute you were right on and the officers should not have been critical. As long as you didn't try to play drill instructor and humiliate the soldier. It may have been that they were day dreaming and never noticed the officers. They just needed to be reminded. They should apologies, salute and go about their business.

You could have said excuse me I noticed you didn’t salute the officer did you not see him

AR600-25 1-5. Says all Army personnel are REQUIRED to salute when they meet and recognize persons entitled to the salute. Further on it states the junior person shall salute first. In my opinion the Major should have kept quiet and appreciated the Specialist's respect and knowledge of military regulations, customs, and courtesy. As. Leader he should abide by and encourage the same.

You did just fine. My preferred method was to yank a knot in their ass. In your scenario to not render a hand salute looks to be intentionally disrespectful. Being so, that PFC is hollering to the two officers F**K YOU for all to hear and you know any other enlisted person that saw the PFC were waiting to see what you will do about it. As for the Majors comments. You just respectfully tell him "ok sir you are telling me that you don't deserve a hand salute, you don't want to be respected. But I will ensure that any other officers that may happen to be in out group WILL receive a proper and respectful hand salute and the greeting of the day."
The same applies the other way. You render a hand salute to an officer and give the proper greeting of the day and the officer does not return your salute. You do an about-face and catch up to the officer, stop him or her, Explain they did not return your salute as required and give another salute. I bet it's returned now.

Ask them if the arm is broken. It worked for me

Plenty have answered the question. I have hit the age where I like to go down memory lane. I have learned there is a big difference between Combat Arms Officers and Non Combat Arms Officers. An Infantry Officer and I (Armor) was tasked to assist a Doctor who was to be commissioned as a Captain. I am not sure how or why it worked out the Doc was there and how Docs get commissioned but we were taking him to clothing sales to get him set up for uniforms and the what nots, before heading to FT Sam Houston. While at the Clothing Sales the other Captain and I noticed a Infantry LTC with pin rank instead of sew on. We convinced the soon to be Captain the LTC had is insignia reversed and it was his duty to correct the LTC. The LTC's uniform was perfect but the Doc did not know that. He was very hesitant however we impressed upon him it would be better he do it than letting him go out of the building and someone else seeing the LTC out of uniform. We stressed, "make sure you tell him you are a Captain and tell him to pay attention to details." Let the LTC know you are a Medical Doctor and he is only Infantry."

It was fun to watch from a distance. The LTC was fairly shocked at first but soon realized the Doc was being played and he handled it like a pro. He realized what was going on and played it perfect. We actually talked to him later and he really had a good laugh. The soon to be Captain didn't talk to us anymore.

So true, any active duty member in uniform should show respect to the rank of others. The only exception I ever saw was when an Admiral started to dress down an E3 in dress uniform for not showing respect. The Admiral had to shut up, pop tall and salute the Seaman when he noticed the MOH on his chest. Never a right or wrong way, it's discipline at stake.

Whack to the Nut sack

I’ve commented before on this. A salute is a sign of respect shared between warriors. While the tradition is for the junior to initiate there is NO Rule that says the senior cannot initiate it. So start the salute - 99.9% of the time the junior realizes it and snaps back a top notch salute. Include a greeting as well like “Good morning soldier” or “All the Way”. And go on about your business. There wasn’t really a problem ... you showed respect to a fellow warrior...and you are both serving this great country. Nuff said.

Just do not do what my driver (a very squared away corporal) did one afternoon while he was driving me trough a training area in an open 1/4 ton (for you young guys, that is what we had before the Humvee and after the M-37 Jeep. Knock it off. I know how old I am. I took basic with the M-1 and was issued brown boots with a bottle of black dye.) when we drove by a group of trainees standing outside of a classroom on a smoke break. I was occupied reading some paperwork when suddenly he slammed on the brakes causing me to nearly eat the windscreen. When I looked up he was already dismounted and halfway to the group of trainees standing about 60 feet away. He bellowed at the top of his lungs attention and got right up in the face of one of the trainees. After about a minute of what appeared to be a by the numbers tail chewing he told the whole group to render a hand salute. After I returned the salute and the corporal had returned to the vehicle. I said, "I must remember to never fail to salute a superior officer when you are around." The corporal in a very excited voice said, "Sir, you did not see what that punk did, he not only did not render you a proper salute, he shot you a peace sign!" I said; "Well, I bet he will never do that again and I bet he is thanking the Lord right now that you were not armed."