Greece had been a part of the Roman empire since long before Galen's time. Does anyone know when all Greeks became Roman citizens? I know it didn't happen throughout the empire until the third century but I'm fairly sure the Greeks were all granted citizenship much earlier.
The Roman empire seems to have been a bit like modern America in that they were Romans but they retained their cultural identity.
If I'm right Galen would have been both a Roman and a Greek.
I suppose there's a distinction to be made between 'ethnicity', which is set once only by one's parents' DNA, and 'nationality', which is an adopted status defined at least once by the political environment. It sounds like he could have been ethnically Greek, but of Roman nationality.
The problem word in the sentence 'Galen was Roman' is the word 'was'. 'To be' always seems to cause problems...