If I'm currently studying to specialize in Urology, for example, is it correct to say, I'm specializing in Urology, or does it imply that I already did my exams and specialized in Urology?
If you've already taken your exams and are a specialist in urology, does that mean that you're a urologist (MD)? If so, then you should say "I'm a urologist".
If you're currently studying to specialize in urology, does that mean that you're a medical student who wants to become a urologist once you graduate from med school, or that you're an MD who's changing specialties from, say, orthopedics to urology? If so, then you should say either "I'm currently studying to specialize in urology" or "I'm studying to become a urologist" (or something similar).
The present progressive "I'm currently studying" means that you're studying urology now; "to specialize in urology" implies that you aren't yet a urologist.