Smith suffered the injury playing for the Cheetahs in February, and informed Springboks coach Peter de Villiers on Wednesday that he was not fit enough for consideration.
De Villiers said: "Juan had been in a race against time to be fit and we were keen to give him every opportunity to prove himself available.
"He is understandably very disappointed - as are we - but he has made the right decision. A fully-fit Juan Smith would have been an enormous asset to our campaign and this is a major disappointment.
"But it was obvious that he was still some way from full fitness and his decision confirms the advice of our medical team. Out of respect for a great Springbok, we wanted to leave the door open until the last moment for him."
Smith, who has featured in 69 Tests for his country, said: "I've worked very hard to get back on the training field but I'm still some way from full fitness and have been training with quite a lot of pain. I'd been trying to tell myself that I'd come right but that wasn't going to happen.
"This is a huge blow to me. The Rugby World Cup is the pinnacle of any player's career and I was looking forward to our attempt to become the first team to win the title in successive tournaments."
The South Africa squad will be named on Tuesday evening.
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