How could I have handled a Soldier displaying gang affiliation on social media better?

SPC (Join to see), there are a lot of things that you can handle at the lowest levels. Gang affiliation really isn't one of them. You did the right thing.

Realistically, there is no "off duty" time when you're in the military. Service members are also bound to a different set of rules (UCMJ among other things). I run into this problem at work where I see my clients (who are also employees of my company) blatantly displaying things that are against company regulations, but as a civilian agency, I can't really enforce something that happens while not on the job site. That is not the case with the military. There are standards of conduct that apply 24/7, and gang affiliation is one of those. In my opinion, you handled it well, by first approaching the individual informally and unofficially with the warning/notification, but when there was no response and no action taken, you acted in accordance with the good of the unit - yet still tried to handle it at a level where there would be no official repercussions to this person's actions by posting such information on Facebook. You sought counsel from more knowledgeable and respected individuals, and followed their advice. This person had multiple opportunities to act on the information you provided and the request, yet still chose not to comply with the code of conduct. I say you did very well in your handling of the situation (and this is from someone who deals with reporting procedures and complaints daily, as I deal with ADA violations, EEOC complaints, and various investigations in my work, all of which have to follow proper handling protocols). The goal is always to resolve an issue at the lowest possible level of consequence unless it's something so huge that such is simply not feasible nor desired. Gang activity was a huge thing when I was in the Army and as such, it is good that you turned it over to CID for them to take action. Just watch your six. Sometimes, legitimately or not, those who are considered to be "whistle blowers" can end up being a target, being seen as a snitch. At this point it's out of your hands, and I'd leave it that way.

Potential criminal activity is best left to those with the task of investigating.
More so when if proven true, the seriousness is much greater.
So, some SM says on FB, my wife left me for some dude, Im staying in housing till I PCS next year.. He might warrant a message saying, dude, look it up, thats all sorts of wrong and can get you in hot water. If he modifys his behavior, problem solved, if he does not and is investigated later, well the paper trail is as damning then as it was before and it's a mostly victimless crime.

If the SM is stating he is dealing, selling stolen property, in a gang, driving drunk every weekend... Thats a note to CID at his home duty station and links to where you observed the statements.