PHOENIX, Ariz. (AP) _ The nation's governors are being urged by Arizona Gov. Bruce Babbitt to denounce apartheid in South Africa and to join him in signing a petition supporting the Free South Africa Movement.

In a letter to the 49 other governors, Babbitt also chastised the Reagan administration for showing ''indifference'' toward South Africa's black majority.

''Most Americans, including citizens of my state, are troubled by apartheid,'' Babbitt said. ''Strong action by the United States can bring about change in South Africa. We must show the entire world that we want change.

Jim West, Babbitt's news secretary, said the letter was prompted by an advertisement in Sunday's New York Times and other newspapers. The ad was paid for by the Free South Africa Movement, which has orchestrated a yearlong protest against apartheid in front of the South African Embassy in Washington.