According to a report by Beeld newspaper on Wednesday, Shadrack Rasello, 26, died in Chris Hani Baragwanath hospital on Sunday, his uncle Lawrence Rasello told the Afrikaans daily.
He had slipped into a coma last week and was being kept on life support machines.
His uncle said the family, who did not want to switch off the machines, was informed by the hospital that he had died.
“They [the hospital] did not tell us how it happened,” said Lawrence Rasello.
His nephew was stuck near the Rustenburg Kloof Resort for three days last month after falling and becoming wedged on a cliff between two large rocks.
More than 100 people from different organisations participated in the rescue, which ended with a surgeon being flown in to amputate his leg.
He had fallen after being frightened at seeing a snake.
His family told Beeld they did not have money to bury him.