You cannot "fire" a member of the military. It is not possible. You can reassign them but they would have to go through a court-martial to be removed from service unless he does it voluntarily.
Pretty much takes a literal act of congress to remove anyone of E7 and above without UCMJ. The Senate Armed Services Committee are the ones who recommend stuff like that. What usually happens is they just aren't promoted and the "up or out" system kicks in and they have to retire. As this is an Office and a LTC at that, he'll be retiring as he just become highly radioactive and military promotions to O6 (COL) and above are super political. HQDA promotes officers to O6 and the Senate promotes to O7 and above (that's generals). The competition for O6 is crazy tough with a long list of requirements, this guy won't be sent to the assignments required to get those requirements.
From the Military's point of view this guy screwed up big, you don't go off on your own like that. Instead your supposed to submit written testimony to HQDA and get permission to testify in front of Congress. That permission is almost always granted, unless the subject itself is classified. If it's about perceived wrongdoing then there are internal channels you go through to report it. LTC Vindman didn't do that and now he's facing the consequences of being relieved of his post. Any assignment he goes to from here is going to keep a lazer and a gag order on him.
Interesting note is what the heck his brother did to be involved in this. The whole "guilty by association" isn't something the Military does, his brother had to of done something wrong to also be relieved. If I had to take a guess I would say it was related to sharing information that shouldn't of been shared.