Smit returned home early from his side's double-header in New Zealand, having sustained the injury during the Springboks 19-8 loss in Wellington last weekend.
And he will now miss the remainder of the competition as well after undergoing surgery.
Despite missing the rest of the tournament, the Springbok captain was upbeat after South Africa's 30-28 win on Saturday earned them their first away win over the All Blacks for 10 years and their first in Carisbrook's "House of Pain" since 1921.
"I have obviously been very down since I returned home, but the guys brought a smile to my face," Smit said.
"It was an incredible win, and vindication for all the hard work that has gone in by all involved.
"The All Blacks are formidable opponents, their home record attests to that, and winning there takes something very special. I am over the moon."
South Africa Rugby revealed no date has yet been set for Smit's return.