Does a U.S. patient have the ability to seek damages for medical malpractice if the surgery occurred in Mexico?

There are not enough facts presented to make that determination. Generally the answer is no. Specifically the facts and circumstances surrounding the connection between the facility and your home state including advertising may create a sufficient nexus for a lawsuit here. A real long shot, however. Even if you could confer jurisdiction collection would be a nightmare.

Maybe. If the surgery took place in Mexico, and the doctors, etc, are only in Mexico, then your only recourse would be to file suit in Mexico, and I have no idea what Mexican law would do. If, however, the doctor was here, either when served or he had an office hear that recruited you to go for the care in Mexico, there might be a case that could be brought here. So, without more facts, I cannot say one way or another. You should consult a lawyer in person.

This is a US forum, so best to ask in a Mexico forum.