ATHENS—Greece has renewed its territorial claims over a broad swath of disputed waters in the eastern Mediterranean where the indebted country hopes to find vast oil and gas deposits—a plan that risks sparking a confrontation with Turkey.

Over the past several weeks, senior government officials have made a series of public statements—both at home and abroad—pointing to an almost two-decade-old international treaty granting those rights, one Greece hasn't asserted until now. Athens also has been building support in other European...