ATHENS—Two years after Europe bailed Greece out to protect the euro, the rescue has become a debacle that threatens to unravel the common currency.

After Greece's May 6 elections left pro-bailout parties too weakened to govern the country, more elections are likely in June, with no guarantee a stable government will emerge. By next month, Athens must identify €11.5 billion, or $15 billion, in fresh spending cuts or face suspension of the international loans it needs to pay...