Thank you. On this website, I do not always get to give good news, and this is one of these times.
There is no law that states that the potential employers cannot share information about the candidate. So unless there was some kind of mutual non-disclosure clause (and likely there was not), they can call another hospital. If this happens, the other hospital has the right to decide to tell them that you may be applying to both, or, not. They chose to admit it.
The only kind of information they cannot share without consent is medical information (HIPAA), and, public disclosure of private facts (which is loosely defined, but normally pertains to that of intimate and domestic nature, not this).
As such, they broke no law. I am afraid that there is nothing you can do but to attempt to salvage the situation the best you can.
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