PIRAEUS, Greece—At a dark crossroads here in September, Greek police kept a safe distance while black-clad activists from the fascist movement Golden Dawn chased and attacked Pavlos Fyssas, a 34-year-old rapper.

The police had long been in the habit of standing by while Golden Dawn's paramilitary squads rolled into action, mirroring the hesitance of Greece's political leadership to deal with the growing movement's muscle. Only after a Golden Dawn member fatally stabbed the rapper did police officers make an arrest, according...