The Rugby Football Union also confirmed that Karl Dickson of Harlequins will join the squad as a replacement.
The news is a huge blow for Youngs, who impressed by scoring two tries in Saturday's 36-27 second Test defeat in Johannesburg.
The 22-year-old Leicester half-back's performance was one of the key factors in England's revival after they fell 28-10 behind early in the second half.
He was withdrawn six minutes from time and replaced by Lee Dickson - Karl's brother - but England gave no indication that he had damaged his right shoulder after the game.
Lee Dickson may now be in pole position to start next Saturday's third Test in Port Elizabeth but Youngs' withdrawal could also offer an opportunity for Danny Care to resurrect his international career.
Care was dropped for the RBS 6 Nations by head coach Stuart Lancaster earlier this year after a series of off-field incidents.
The 25-year-old impressed in an England XV's victory over a Southern Barbarians side in Kimberley last week and could now be ready for elevation to the senior side.
Care's Quins team-mate Karl Dickson, 29, is due to fly Johannesburg overnight and will immediately be in consideration for Tuesday's tour match against the Northern Barbarians in Potchefstroom.
He has been summoned a day after Jonny May of Gloucester was called up to replace injured Harlequins wing George Lowe in the squad.
Bath's Nick Abendanon last week arrived after injury to Quins full-back Mike Brown while London Irish flanker Jamie Gibson replaced London Wasps' Joe Launchbury before the first Test.
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