Most of the oath has no enforceable criminal law counterparts. In a number of US states, there is enforcement of the promise that "I will not give to a woman a pessary to cause abortion". Also in most states, "Neither will I administer a poison to anybody when asked to do so, nor will I suggest such a course", though there would need to be discussion of what a "poison" is. If we assume that the clause "I will abstain from all intentional wrong-doing and harm" is part of the original oath, that would be criminally or civilly enforceable, even against non-doctors. The obligation to teach for free is definitively not enforceable.
However, there do exist "medical ethics", which tell doctors things that then must or must not do. The HIPAA privacy rule is one of thousand of such enforceable requirements. A doctor has a special duty of care to his patients that goes beyond ordinary person-to-person duty of care, and to the extent that some code of ethics prohibits an action by a doctor (and the doctor harms the patient), the doctor can be sued. A further action for violation of medical ethics would be suspension or revocation of the license to practice medicine.
Since you're apparently looking beyond civil actions, I assume you are referring to criminal prohibitions, but also criminal prohibitions of actions by doctors (not general "don't rape" prohibitions). Laws against abortion are directed against doctors, though they are written with the presumption that a non-doctor could never perform an abortion. Also, negligence (by anyone) could rise to the level of being a crime, e.g. in Washington "A person is criminally negligent or acts with criminal negligence when he or she fails to be aware of a substantial risk that a wrongful act may occur and his or her failure to be aware of such substantial risk constitutes a gross deviation from the standard of care that a reasonable person would exercise in the same situation".
There are numerous things that only doctors can do, such as prescribing medicines. There are, correspondingly, prohibitions against doing those things outside of legally-prescribed boundaries, they can be prosecuted, but again prosecution of a doctor for "running a drug mill" is for violation of 21 USC 841, which everyone can be prosecuted for. As far as I know, there are no actions which are criminally prosecutable and illegal for a doctor – and no one else – to undertake.
