Cronje, who started the first and second test wins, has displayed delayed symptoms of concussion from the 37-15 victory in Durban on Saturday.
Springbok coach Allister Coetzee, who has cover in the form of Francois Hougaard and Rudy Paige, will not name a replacement, South Africa Rugby said on Tuesday.
Meanwhile, Coetzee has released centres Lionel Mapoe and Francois Venter to the South African A side that will take on the French Barbarians at the Orlando Stadium on Friday.
That means Jesse Kriel, who sustained concussion in the 37-14 first test win over France, is set to regain his place at outside centre having been passed fit.
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Coetzee will name his side on Thursday.
(Reporting by Nick Said, editing by Ed Osmond)
