If a plastic surgeon in Manhattan, NY won't refund deposit paid for a canceled plastic surgery operation, what are my rights?

The first thing to do is to check the terms and conditions, as reflected in any contracts or agreements you signed, and/or prominently on any marketing or advertising materials, about cancellations and refunds. If the terms and conditions address the issue of cancellations, they will control--so if they say the deposit was nonrefundable, the surgeon may keep it and return the rest; if it says the full amount is nonrefundable under the circumstances under which you cancelled (e.g. 1 week notice), then the surgeon can in fact keep it.

If there is nothing about a cancellation policy at all, it would most likely be the case that the surgeon could not keep the full amount--in commerce generally, the vendor (the surgeon) is only entitled to the money if he/she provides the good or service, so if somone does not purchase, they don't have  to pay. It would as you note be reasonable for the surgeon to keep the deposit; that's the general purpose of a deposit, and it could easily be imputed to this situation even if the contrat, agreement, etc. was silent on the subject.

You may wish to take any paperwork, etc. to an attorney who can review it for you--and represent you against the surgeon, if need be.

This matter is covered by contract law.  If you have a contract with the doctor relating to the services that were going to be performed, there may be a clause in the document that the doctor's office is permitted to keep its money in the event that the procedure does not go forward.  The doctor keeps his schedule open for this time and plans on having certain personnel available that he has to pay to be there.  Thus, the doctor may have the non-refundable policy based on his expenses to book the OR and hire the people to be present.  I suggest that you show the contract or any documents, such as hospital polices to a lawyer to determine whether the doctor may keep this money as it is a lot and potentially an unjust enrichment for the doctor.  You may have  a claim against the doctor depending on the contract since he did not perform the services.