How should I handle being a new Team Leader when my predecessor is making inappropriate comments about me, both online and in the workplace?

Ignore him. He's out the door. Your new team members will make their own impression of you as soon as he's gone. One day down the road, you'll probably outrank him and you can enjoy the sweet irony of it all

I say bury him. Document what you have. Take screen shots of his on-line posts and rantings. Line up your ducks for an EO complaint and definitively end him.

I may be going against the grains of most the comments here, but too often I've seen the "it shall pass", or "they are getting out anyway", or "just let it be" as the major source of continuous problems of moral and working environment. Letting this go in this regard is a continuation of that.

There are two lines of thought about short timers.
1) They can do what they want because they are moving on soon and everyone will ignore them.
2) They will tow the line their final days because the system will hem them up with last minute negative administrative actions that will delay them moving on.

I prefer the later...........

Hell, skip your Chain of Command and go straight to IG. Then what will happen is it will flow back DOWN to your 1SG but not before IG went to the Company Commander.

No one is going to protect the career of a Corporal from a Specialist. What sucks for him is now this could be documented on an NCOER.

Don't let this sit, because someone like me is depending on you to speak up. Otherwise someone like me runs the risk of his career getting a black mark if actions like this turn into something else when there were warning signs all around.

Do not let this roll get all of your information from Face book and statements from your team and pass it up the COC. People like this young CPL are what causes Toxic leadership to breed and expand, let the COC deal with him/her. MTC