Critics say Chavez uses the term "Bolivarian" to describe his political movement, and some believe his government intends to win over hearts and minds through classroom indoctrination.
Chavez says the law is necessary to change Venezuela's "bourgeois" educational system.
Tareck El Aissami, Venezuela's justice minister, accused the protesters of "inciting violence" by throwing rocks and other objects at police.
Health authorities said they treated dozens of people for tear-gas inhalation and at least 14 who were hit by rubber bullets or displayed other minor injuries.
Juan Francisco Romero, the interior vice-minister, said at least a dozen police were mildly injured.
Pro-government legislators deny the law aims at political indoctrination.