When an E-4 calls me "hey" or "hey man" knowing that I am an officer, should I bother to correct him?

As an infantry officer, you should absolutely correct that deficiency. My recommendation, as an NCO, would be to correct him on the spot (with tact) then talk to his first-line (team/squad leader) to allow them to provide further correction. I can't stand when officers correct my soldiers for me and they likely won't either.

With that said, anyone, regardless of rank, should take it upon themselves to correct any deficiency. As long as it is done with respect and tact, it should be well received. "Check down, not up" is not an acceptable response.

Absolutely!!! You earned that rank. If don’t then you are failing that Soldier and that soldier will do it to other officers. You are doing a favor for that soldier by correcting him or her because there are other officers and even non commissioned officers that will tear that soldier apart for not properly addressing an officer. The other problem that could exist if you don’t correct that is other Soldiers may see that kind of behavior and mimic it. Officers and NCOs should not let things like this slide.

Sir, if he adtessed you as "sir" during duty hours, but referes to you as "hey man" outside of work. I would not correct him. He/she obviously distinguishes work from personal life. At the end of the day were all human, we all bleed red, and we all fly the same flag. If the soldier does it out of work its fine, maybe even get to know him some. As a SPC in yhe S6, most of the officers I k ow outside of work (and in work) are cool, because they know ill help them.

Correct him or you have established a new standard that you will have to live with

Yes. Politely and privately correct him.

It would be wrong (too lax) to not correct the E-4. I know things have changed since I was in but Officer/Enlisted decorium should be maintained.

Captain, respect, and discipline are required for obvious reasons. Some "Speedy 4" is not my drinking buddy, and in my case, neither is a 3rd Class PO. I not only would correct him/her, but I'd also jack them up so far up the mast they'd think they were a pennant. Good order and discipline are necessary and required. Do not let a situation like this slide otherwise the subordinate will begin to think that they can let other things slide and that kind of thinking may cost some folks their lives,

Yeah, need to correct that, politely. My favorite comment to enlisted that didn't salute or call me sir: "When did you decide that I didn't earn my commission?"

Tough one, if it’s outside work but you want to maintain a professional relationship or professional footing then I’m sure all you have to do is ask, I’d say “hey Specialist, let’s keep it professional so I’d appreciate it if you refer to me as sir or (rank)”

OK, I feel like the Mad Cop on YouTube lol. I know this has to be a joke and you’re just posting this on here for fun. That being said if you even need to post this question then I will tell you to finish your duties for the day and or week depending on what you have going on while you have free time go ahead and write your letter resigning your commission because you were the kind of officer that I really infuriated me when I was enlisted as well as when I became an officer. If you think that you should not correct this soldier or airman or marine or whatever the case may be and then you your self or part of the problem, and will cause this problem to continue because you don’t fix it. I’ll tell you something I learned a long time ago by NCOs, respect is earned not just given regardless of what anybody says.
I will say that as an officer, primarily as a lieutenant because I came from a transportation company and then volunteered and deployed with an aviation battalion and then came home and volunteered with inventory that I had a different mindset than others. If we were working out in the field my rank is just a title running up and saluting me and service and serve that when the in-state is to complete mission, not to show who the most courteous soldier is, so often times I would be more relaxed with certain customs and courtesies unless there was a higher ranking officer around then do what you’re supposed to do. Another reason I was like this is because a long time ago I had a talk from a MAJ That told me in other words, you need to decide whether you’re going to be enlisted or an officer because you cannot be both. If you just let a man or hey continuously slide then you have chosen to be enlisted and not an NCO but lower and listed and I’m not saying that there’s anything wrong with them because I worked my way up from an E-1 to a Company Commander then BN staff. Another way you can look at this which is what I still do to this day even retired out of the military is back when we had respect for uniform and had to shine or boots etc. Soldiers that could not do a simplistic task such as shine their boots, you have to ask yourself what else will they or won’t they do when the time comes. And please understand I use the expression when the time comes very probably but it can apply to anything, but in your particular situation it applies to the fact that if I person(s) continuously calls you incorrectly and that’s just something as simple as changing a few words that come out of their mouth, then what in the world are they doing behind the scenes that you’re not aware of that they’re being told to do but not only you but they’re in CEOs or when it comes time to go to the field or when you deploy in whatever capacity that may be.
I know this seems, who am I line to, I know it is a long message about somebody saying hey man, but you have to understand that regardless of what branch of the service you were in there are customs and courtesies they go way back and they are in place for a reason and yes we, you, the next person do not uphold those courtesies then and I’ll actuality it’s you yourself that are doing a disservice to that soldier as well as the service as a whole because you did not correct an action or behavior which with something within your lane that you could take care of and you didn’t and this will continue on.
One last thing, not sure about you but I busted my ass to become an officer, hell I even volunteered to go to Iraq just to make sure that I would not be light on the right when I became an officer as well as gave me an opportunity to watch good and bad officers and learn from them. Now that I have said my piece I will let the NCOs do what they do best, lead and probably beg to know who the soldier is praying to God that they know them so they can crush their soul lol.