CAIRO—Egyptian President Hosni Mubarak handed power to his vice president but retained his title, a half-measure that confused observers, angered opposition leaders and provoked an uproar from hundreds of thousands of protesters massed in the center of the country's capital.

The move sets up another conflict with the opposition movement, which has called on supporters to gather for a huge protest Friday. Expectations that the president would resign had built through the day, and the immediate reaction to the speech was anger,...