The Lions number eight led the Springboks to victory over France in the first two Tests after being named as captain by Allister Coetzee last month.
Whiteley missed the win over Les Bleus in the third Test last Saturday, though, after doing damage during the warm-up at Ellis Park.
Coetzee revealed after his side sealed a whitewash over Guy Noves' men that his skipper would discover the extent of his injury on Monday and it was confirmed on Tuesday that he faces a lengthy spell on the sidelines with damaged ligaments.
"We expect to rehabilitate him in six to eight weeks," Lions team doctor Rob Collins said.
South Africa start their Rugby Championship campaign against Argentina in Port Elizabeth on August 19.