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Your psychiatrist friend is still new in the field of medicine and probably doesn't realize the gravity of what she has done. Legally speaking, since she is technically not your psychiatrist, she did not have the legal obligation to protect your privacy. What she did what hurtful and unprofessional, but not illegal.
Speaking with her clerkship director is not a bad idea. He/she needs to educate the residents so they do not make the same mistake again. Making psychiatric diagnosis in a casual manner can significantly damage another person's reputation.
I hope this is helpful.
I totally agree with you. What she did was wrong and unprofessional. Unfortunately this happens very often. She should be told not to do this, but to prosecute her in a court of law based on HIPAA violation is difficult because you are not her patient. She has no medical record of you.
Regarding the slander issue, I suppose you can try that angle. However, I do think that your idea of speaking to her clerkship director is a great one. Afterwards you can consider other options.