Howdy! I applied to health communication doctoral programs this year, and let me tell you -- it can be challenging to find programs that brand themselves as health comm. Case in point -- I actively started my search nine months before the deadlines, and I still found a new program the night before the application deadline (still wish I could've applied, UT Austin!).
A great starting place is NCA's Doctoral Program Guide. NCA, or the National Communication Association, compiles a doctoral guide which is searchable by research specialty. It's particularly helpful because you can get a pretty good ~vibe~ of where the active research is happening. In fact, I think that this will be the most helpful resource for you because you can search for both intercultural and health communication programs (social networks is a little more niche: I've seen social network research done in health communication, social media, comm & tech, etc.). The NCA guide will also give you admissions requirements, research emphases, # of communication doctorates conferred, names of active professors, etc.
Before you dive too deep into the guide, you need to know what kind of communication you're interested in. Comm is quite broad, and largely falls into two different camps: rhetoric and social behavioral sciences. Sometimes you'll have departments with both contingents, but in my experience, you'll find these programs in separate depts. I don't have much experience in the rhetoric camp, but in social behavioral sciences, you'll also have to ask yourself about methodology -- are you more of a qualitative or quantitative person? Finally, are you interested in interpersonal communication (person-to-person) or in mass communication (advertising, news, media, etc.). If you know what you're interested broadly (e.g. interpersonal communication), then you can first search for interpersonal communication programs, and see if there are any people that work on health stuff. I'm really interested in mass media effects and advertising (in the context of health) so once I realized that the NCA was missing integral programs, I broadened my search to mass comm and was able to find some really interesting programs. Starting broad is integral because in my experience, not many professors consider themselves health comm scholars. If you're sticking to "health comm" google searches, you're going to miss some stuff.
One final caveat: An (unfortunate) number of these programs require a master's.
Here's a list of schools with health communication research from NCA. I encourage you to check out the website -- it has some good info!
Bowling Green State
Texas A & M
Other notable programs that you won't find on the NCA website:
Northwestern: Strong in health communication. Rumor is that they are going to create a health communication PhD program in the next couple of years.
UC Santa Barbara: Have more of a focus in interpersonal communication than those that are listed here, but does have a couple of faculty that conduct health communication research.
UNC Chapel Hill: Strong in health communication, but is unfortunately only a three-year PhD program.
UT Austin: Offers a PhD in "advertising" instead of communication, but it's very strong in health communication. Probably one of the strongest "health comm" programs out of all of those listed; most of the graduate students were studying health (or something closely related, such as risk).
Wisconsin-Madison: It doesn't strictly have a *health communication* contingent, but it's a strong, well-funded department and one of the most prolific health communication scholars, Dr. Dhavan Shah.
Feel free to PM me any questions!! I got my B.S. in communication and currently work as a lab manager in a communication department (and ya know, just went through the app process), so I have some knowledge to share!