PARIS—France will close embassies and French schools in 20 countries on Friday, its Foreign Ministry said, amid fears of a backlash after a French magazine published a series of caricatures of the Prophet Muhammad.

French authorities said they feared the cartoons published Wednesday in satirical magazine Charlie Hebdo could cause more outrage in the Muslim world, days after a video denigrating the Prophet Muhammad helped to fuel violent protests at U.S. and other Western embassies in several Muslim...