"Firstly, it's really tough on Matt but he has been assessed by the medics and he wouldn't be able to train or play for some time," said England head coach Stuart Lancaster.
"We wish him all the best but it's a great opportunity for Rupert, who has been the first-choice tight head for Gloucester this season and also involved with the Saxons during the Six Nations."
Stevens, who has made six appearances for the Lions, is the latest casualty to miss the three-match series against the Springboks.
Fellow Lion Tom Croft and Tom Wood have both been ruled out of the tour, while Toby Flood and Phil Dowson are also battling injuries.
England begin their series with the Springboks on June 9, in Durban. They then face the Lions' last opponents in Johannesburg on June 16, before the final game on June 23, in Port Elizabeth.