The 32-year-old had missed the matches against Wales and Samoa, but was hopeful to play in this weekend's quarter-final against Australia.
"I'm very disappointed after all the hard work that has been put in over the past few months, but it's not about me, it's about the team now and there's still a big job to do," Botha said.
"I'll be sitting at home with my family on Sunday with a tear on my cheek as the team takes on the Wallabies.
"But we can't stop here - there are still three games left to play."
This could be the end of an international career for Botha, who will join Toulon at the end of the tournament.