Rabbi of Berlin, the driver does melachot d’Oraita.
Cherybim, we cannot learn halachot from a mmaaseh rav (Baba Batra 130b). You do not know the reason why the “chosuv rav” thought that it was permissible to him (nor why the doctor drove himself instead of calling an ambulance driven by a goy, which is the preferred method – and, in fact, the nurses in my community ride to and from the hospital in a clearly marked van driven by someone who is not halachically Jewish).