JOHANNESBURG—South African President Jacob Zuma, reacting to a firestorm of criticism, has apologized for fathering a child with a woman who wasn't one of his wives.

In a statement Saturday, the 67-year-old president said the birth has put pressure on his family and his political organization, the African National Congress. He also acknowledged the anger of South Africans over the matter, saying, "I deeply regret the pain that I have caused." the president said in the statement.

Mr....