The 29-year-old De Zorzi, who has 17 test caps, suffered the injury while fielding during the last one-day international against Australia in Potchefstroom on Wednesday.
"Subsequent scans on Friday revealed the extent of the injury. No replacement has been named," CSA said in a statement.
South Africa will go into the test, which starts in Durban on Friday, having suffered several hamstring injuries which have ruled out fast bowlers Ottneil Baartman, Kwena Maphaka and Lungi Ngidi and left leading paceman Kagiso Rabada doubtful to play.
Rabada has been included in a preliminary squad, but his availability for the first test will be assessed after an intra-squad match on Monday and Tuesday.