As mentioned above, psychology licensure requirements are dictated by the state in which you wish to be licensed. In MA, where I am licensed, you could get licensed with a degree in applied social psych. You would still have to meet specific coursework and supervision criteria, and these are, however, more likely to be a required part of a Clinical or Counseling program than of a Social Psych. program. If you want to provide psychotherapeutic services as a licensed psychologist in MA, you will have to have a additional special designation with your psychologist license know as "Health Service Provider" (HSP). This designation requires, among other things, that you complete a internship that is APA accredited or meets very similar minimum criteria. It would be difficult to get such an internship coming from a Social Psych. doctoral program.
If you're curious, the language from the MA licensure application requires you to state your area of training and experience as follows (from the application):
"21. Identify the field(s) within Psychology for which you have obtained your doctoral degree and supervised training. This is the field(s) in which you will be qualified to practice. The field of
your doctoral degree is usually printed on your transcript, e.g. "Ph.D., Counseling Psychology".
Use the following fields to complete this item. If none of these areas fit your training and
experience, describe your own field in completing this item.
Behavioral Psychology
Experimental Psychology
Psycholinguistics
Clinical Psychology
Forensic Psychology
Psychometrics
Community Psychology
Mathematical Psychology
Psychopharmacology
Consulting Psychology
Military Psychology
School Psychology
Counseling Psychology
Neuropsychology
Social Psychology
Developmental Psychology
Personality Psychology
Educational Psychology
Physiological Psychology"
