Could I be kicked out for rotator cuff injury?

Always be careful about posting medical questions as the readership is large. In general, always go to the source, not urban legends, what people think they've seen, etc. There's a DoD and Army version or repub of disqualifying conditions. Most of them which deal with ortho stuff typically refer to "untreatable" or "loss of this much range of motion", etc. So if you're not in one of those categories (won't know until you're evaluated), recovery is just a process. What they don't like is someone who has an issue, keeping their mouth shut, and then bringing it up while deployed. Then your are a larger burden on the system. You don't want the issue to become something other than medical. Make sure you get a LOD determination and keep a copy of it in your safety deposit box. They frequently show up missing later on when you're working the VA application.

Regardless of how it could turn out; it is your responsibility to inform your command of your status. It directly effects mission readiness; they make decision based off personnel status. So I am in agreement with SPC Delgadillo that you're hurting yourself but also essentially you're hurting the unit (2nd and 3rd order effect). I do understand your hesitation and the reason for that but still you should report it.

Why would you want to hide a medical issue?