The Manchester United defender was injured in a tackle during England’s first training session at their base in Rustenburg.
The 31-year-old Red Devil started just 13 Premier League games last season and the news is a huge blow to manager Fabio Capello as his side prepare to face the USA in eight days time.
A scan arranged by the England medical staff immediately after training on Friday afternoon revealed the extent of the injury, which will rule the Old Trafford star out of action for four-six weeks.
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Tottenham Hotspur centre-back Michael Dawson will fly to South Africa on Friday evening and join up with the rest of the squad over the weekend.
The Spurs man was in impressive form as the White Hart Lane club finished the 2009/10 season in fourth place in the Premier League, sealing Champions League football for the first time in Tottenham's history.
Dawson, currently uncapped at senior level, was a member of Capello's provisional 30 man squad for the World Cup, but initially failed to make the Italian's final 23 party on Tuesday afternoon.
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Liverpool midfielder Steven Gerrard, meanwhile, will now lead the England team as captain in South Africa.
Speaking of the day's events, Capello said: "It is obviously bad news and everyone with the squad is very disappointed and sorry for Rio.
"It was an accidental injury in training, but had nothing to do with the pitch."
