ATHENS—Greek Prime Minister Alexis Tsipras resigned in a bid to trigger snap elections and return to power stronger, plunging his country into weeks of political paralysis just as it seemed to have scraped through a summer of fraught bailout talks and near-bankruptcy.

Mr. Tsipras’s gambit, announced in a short televised address late on Thursday, is expected to lead to his re-election in September, thanks to his popularity and the absence of strong challengers. But whether elections bolster his grip on Parliament, as he hopes,...