He could very well have kept his mouth shut and pitched his way back in the rotation over the next few weeks. Based on how others have performed it seems likely. By forcing the situation at his own desired pace it does nothing but undermine team chemistry.
If he refuses to pitch in relief they could suspend him and not pay him and that gets complicated. I just have trouble believing that will be the course of action. At a minimum he's probably on waivers to see if there is another team interested but we all know that's unlikely.
The lack of a public response from the team makes me think they are looking for him to apologize and move on, or most likely, looking for the cheapest way to exit the situation.
If they succumb to his desire to return to the rotation I'd have serious issues as a player on this team.