The 33-year-old damaged himself in the first Test match against Australia at the Waca in Perth last week.
Proteas team manager Dr Mohammed Moosajee says: “Dale had successful surgery on Thursday where the fracture in the right shoulder blade was fixated with the placement of a screw.
“We expect a minimum six-month rehabilitation period before he can start any form of bowling.
“It is crucial to give the injury enough time to heal and to make sure that he is completely pain-free before he starts bowling again.
“He will begin with his rehabilitation programme once the satisfactory bony healing has taken place; this is excepted to be in a few weeks.
“We are positive that he will make a successful return to international cricket.”