You actually raise a few issues.
First, breach of health privacy is governed by federal law under the Health Insurance Portability and Accountability Act (HIPAA). There is no civil lawsuit provision to this federal law. Instead, violations are investigated by the Office of Civil Rights in the Department of Health and Human Services (http://www.hhs.gov/ocr/privacy/).
Additionally, you can file a complaint with your State Board of Nursing. This is not a lawsuit, but is a complaint seeking disciplinary action against the nurse for breaking state and/or federal rules regarding patient confidentiality and privacy.
You probably want to notify the physician as well (if the nurse is employed by the doctor), as breach of patient confidentiality is a serious issue.
Depending on your State's law, you may also have a legal claim for damages for "invasion of privacy." In that those States which continue to recognize this tort claim, the victim generally has to prove that the defendant published private facts about an individual which portrayed that victim in a false light or exposed extremely sensitive information causing distress.
Similarly, depending on your State's law, you may also have a legal claim for damages for "infliction of emotional distress." These are difficult claims to prove, and generally require severe emotional distress (i.e. proof that you sought treatment from a mental health professional because of the distress).
You should check with a lawyer in your area immediately, as the statute of limitations are very short for these types of actions (many States limit the time period to sue to less than one (1) year from the date of the incident).
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