He is set to fly back to South Africa on Sunday for further treatment.
Oosthuizen missed the last three games of the Rugby Championship having also sat out the whole of 2016 because of a neck injury.
A decision on a tour replacement for Oosthuizen, 28, is to be made later, the union added.
Following the defeat to Ireland, the Springboks take on France in Paris on Saturday, before tests in Italy and Wales.
“We lacked patience in our build up and got off to a terrible start when we lost Coenie Oosthuizen in the first minute of the game, and then conceded two penalties at the set piece,” said coach Allister Coetzee on Sunday.
“Our discipline let us down, it allowed them to build scoreboard pressure and then it becomes difficult to play catch up rugby. We created chances but our execution was not good while Ireland were clinical.
“We must now focus on France and work hard in the coming week to rectify things,” he added.
(Reporting by Mark Gleeson; Editing by Toby Davis)
