I am finding out that my ex-wife may have kept from me a genetic predisposition to Down syndrome and other abnormalities in the baby. The physician did not tell me about this either. I want to know what were my rights as the father. Did they have an obligation to disclose this information?
The regulations known as HIPAA are here. A good place to start is with who the regulations apply to. §160.10 lays that out: a health plan, health care clearinghouse, care provider, their business associates, and the Inspector General. Family members are not regulated entities. Subpart E regulates individually identifiable health information (such as "person X has condition Y") – this is protected information. §164.502 states that
A covered entity or business associate may not use or disclose protected health information except as permitted or required by this subpart or by subpart C of part 160 of this subchapter.
The first permitted use / disclosure is that the information may be revealed to the individual (the individual). Under 164.502(a)(2)(1) they are required to reveal information to an individual "when requested under, and required by §164.524 or §164.528" (you have a right to see your own record; you have a right to know who knows). 164.502(g) requires the covered entity to treat the "Personal Representative" as the same as the individual, and then there is a list of conditions surrounding being a personal representative, reducing to being empowered under law to act on behalf of the individual in making health decisions (it gets complicated in dealing with minors). 164.510 covers disclosure in emergencies, and under (b)(2) allows disclosure to a family member with advance consent of the individual. And so on. There is no general rule "and the covered entity may / must reveal private information to an interested individual", even a spouse, without consent of the individual.
In other words, the doctor has an obligation to not disclose this information, unless permitted by the individual.