ATHENS—In the depths of Greece’s economic crisis, members of the fascist movement Golden Dawn brazenly assaulted immigrants, political opponents and labor unionists. After police said a member killed a left-wing rap artist in front of witnesses and confessed, Greek authorities acted, prosecuting the party as a criminal organization.

Nearly six years later, the trial of Golden Dawn has stalled in a case that highlights Greece’s dysfunctional justice system. The party, known for its stiff-armed salutes, swastika-like flag and...