It is not the responsibility of the ER, in most cases, to diagnose the problem. The ER's responsibility is to see that the patient continues to breathe and that their heart continues to beat.
Diagnostic work is, again in most cases, the responsibility of the PCP or specialist who provides the follow up care once the patient has been stabilized. While there can be some situation specific exceptions to this, I see nothing in the OP's account that suggests that any such exception exists.
Two things I am tired of typing: 1.) A wrongful termination does not mean that you were fired for something you didn't do; it means that you were fired for a reason prohibited by law. 2.) The above answer, whatever it is, assumes that no legally binding contract or CBA expressly says otherwise. If it does, the terms of the contract apply.