Based on everything you described, I think you did an EXCELLENT job. I imagine that I probably would have done precisely the same things that you did.
I'm not sure I see what, if anything, you actually did wrong. From the way you describe, you were not "stopping the RN from going home" - you were simply providing her with information about this patient.
It's interesting, btw, that this nurse's response was "I'll let the new RN deal with it". I wonder what your nursing instructor would have thought if you had said "Well, this isn't my patient, so I'll this let patient's student nurse deal with it." I'm not saying the outgoing-shift nurse was saying or doing anything wrong by any means, but still.
I'm also wondering if by chance the doctor didn't pull either your instructor and/or the charge nurse aside and ask "Why did you have a student - who cannot take phone orders - page me?" Your instructor and/or the charge nurse may have then felt a bit embarassed, and hence in the heat of the moment your instructor pulled you aside and said what he said to you. Which, if so, I'm not saying anybody did anything wrong here; it's just that some petty emotions flared up a bit, and you happened to get caught in the crossfire.
I imagine that if you had NOT taken the actions that you did, you could have just as easily been pulled aside and reprimanded for not having done enough. And that kind of reprimand may have been even worse of an unloading on you.
I'm also wondering why the new staff didn't simply take the stairs.
In any event, what if, hypothetically, this patient coded and died? Would you have been held responsible for anything in court? I doubt it, based on your actions. The hospital, though, probably would have been in SERIOUS trouble, esp since you had done them the silly inconvenience of actually alerting them to a problem with one of the patients. Whereas, had you NOT spoken up, the hospital probably could have legitimately claimed that they were not aware of the patient's status and/or unable to adequately address it, and hence they would be off the hook. The result of THAT would have probably been good for the hospital...probably not so good for the patient and family, though.
I think you did all you could. I honestly can't imagine what else you could have possibly done. I'm just a student like you, so maybe I'm missing something here, but at the very least I simply can't imagine you "relaxing" or simply doing nothing in a situation like this would fly over well on a test question.
I understand that as a SN you are not an employee, and there was probably a world of finer legal concerns at work here. But the thing is, you went to people who WERE employees (nurse, charge nurse, etc.) and you complied with what they asked you to do.