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

Definitely. You are an officer, and he is enlisted... end of story.

You are his superior 24/7. Whether you work together or not. Nip it in the bud, before it gets worse.

It bothers me that you need to ask this question sir. Address this type of issue directly to the offender, and do it the moment it happens, unless you are in the middle of serious operations. Are there other enlisted people that you allow to address you in an unprofessional and disrespectful manner because you are friends? Officers should never fraternize with enlisted, and you should be "an officer" all of the time on and off duty. Either this enlisted person has a problem, or your command has a problem somewhere in the officer ranks.

You have two options in my opinion based on my experience as a Senior enlisted leader and without more details you'll have to decide Sir. Let me say that absolutely you need to gig this Specialist. His behavior is incorrect, disrespectful and he needs to get a course correction before he passes his bad habits on. If you know his reporting senior you can go that route and let them handle what for me would be an embarrassing decficiency in my Soldier (Sailor in my case). That would be ideal, but if he does it again and you don't know the senior you need to correct him in firm but not abusive manner if others are present. If you are alone feel free to light him up a little bit. You earned the right to his respect regardless of your paygrade. I'd be lit up if he did this to a Sgt much less and "O". His Sgt if he's worth his salt will appreciate it. Make him a good leader by making him a good follower.

Correct him, period!

Well being an Officer, I would think that you you call him over to you and bring him to Parade Rest and explain the consequences of In-Subordination and how you could make him a PFC. That should gain his respect. Todays Army is like non before. The discipline is really lax and this is one reason why.

You're supposedly an O-3 and you don't know the answer to this question??

Do you expect every enlisted soldier on your post to know you and address you by rank out of uniform? Because if you expect the SPC who isn’t under your command to address you as such, then that is the standard. Good luck identifying all the out of uniform enlisted soldiers on your next Walmart trip, because standards must be enforced uniformly. You should check yourself; if you keep running into this specific SPC, maybe you need to adjust your personal life to avoid fraternizing with enlisted soldiers. The rank stops when you put on Levi’s. And if you tried to enforce that bullshit on me off post, I’d call the local police on you for harassment. Truth.

Not to correct him will result in a degradation of the good order that is a military necessity.

Had a joint medical assignment with Army-Air Force. Enlisted Air Force medical personnel addressing our Army Doctors, officers all, with "Yo, Buddy" and "Hey." Met with them privately, at attention, told them "No more." The following week they began calling the medical officers by their first name. The Army Doctors told them, "When in Rome, do as the Roman's, call me by my first name." Full bird Colonel. Air Force Enlisted of all ranks hollering down the hall, "Hey Bob, does he need an IV? Army Medics said "No Thank you," and continued to refer to the OIC as Sir. The culture at that assignment was a real challenge.