If a soldier has suicidal thoughts while on deployment, do they get med boarded and kicked out?

Not necessarily, in fact, not likely.
What needs to happen is that Soldier need to come off the line, talk to a professional, and decompress. Once the stressors have been relieved or dealt with, it is probable that Soldier can go back on the line and do his/her duty.
If they really have issues of a magnitude that does require a med board, they should be identified and given exactly that. It is what is best for everyone - the Soldier, the unit, and the Army.

If all the troops that get suicidal thoughts get med boarded and kicked out, that would probably eliminate 1/3 to 1/2 of serving personnel, IMHO. I had suicidal thoughts when I got back from Vietnam, but I dealt with them on my own and told nobody, as that wasn't good for one's profile, and I got over it. IF a servicman/woman has suicidal thoughts and thinks enough to tell somebody about them, I don't think they should have a stigma cast upon themselves, and their leadership should take the matter in hand through personal counselling, supervision, keeping a close watch on them, and (if necessary) getting the troop to therapy. If a troop tells somebody about their suicidal thoughts, I see that as a cry for help, and they should get it. Only as a last resort should a troop get boarded and discharged, after leadership has done all it can possibly do. If a troop tells nobody, then they're either dealing with it on their own because they view a confession as a sign of mental weakness, or they intend to go through with it (in which case there's nothing anybody can do about it). Usually, the ones who intend to go through with a suicide attempt tell nobody ahead of time. When I was a DI, we had a couple of recruits that attempted to slash their wrist with a Track 2 razorblade cartridge, but failed (did it the wrong direction) - I saw that as a cry for help because they were too mentally weak to handle the stress, and the recruits were immediately sent to sick bay to make contact with a psychologist. That's just my two cents worth - a medical professional might have a different view.

I would first focus on getting that Soldier to COSC STAT and worry about a med board later (if it even comes to that).