TBILISI, Georgia—Georgia's parliament approved a new constitution Friday that critics say paves the way for President Mikheil Saakashvili to retain power as prime minister after he steps down as president, much as his arch-rival Vladimir Putin did in Russia.

The changes curtail the powers of Georgia's president and boost those of the prime minister. Critically, the president will no longer be able to dismiss the government, while the prime minister will now head the executive branch, taking responsibility for naming cabinet...