A day of political shock waves in South Africa put new pressure on President Jacob Zuma, as an official report alleged that one of the country’s richest families influenced cabinet appointments and thousands took to the streets of Pretoria to demand his resignation before his second term ends in 2019.

Commentators called the report the biggest political scandal since South Africa’s transition to democracy and said the allegations, if confirmed, could mark a turning point in a brutal leadership battle within the ruling African...