I don't think she can do this without causing big trouble to her

Even police cannot ask if a person resides in US illegally, who gave her the right to do so? If she does that, she violates your right of privacy. You gave her your contact info and etc. to get medical help. She does not have any right to use it in anyway other than facilitate the treatment (or get reimbursed for medical spending). Ask for a copy of "patient's right of privacy" from the hospital. (Also, Google it.) Talk to hospital administration to complain if needed.