Recalled for the second Test in Bloemfontein last weekend, he lasted only 17 minutes before being replaced by Josh Adams, who set up a dramatic 13-12 victory by scoring a late try.
The WRU said no replacement will be flown to South Africa, with Adams set to be included when the team is named on Thursday.
There are fitness doubts over captain and flyhalf Dan Bigger, another victim of a second Test shoulder injury, and tighthead prop Dillon Lewis, who left the field with an issue and later had an arm in a sling, for Saturday’s decider.
Gareth Anscombe, whose 79th-minute touchline conversion won the Bloemfontein Test, will probably start if Biggar cannot play.
Should Lewis be ruled out, Sam Wainwright, who made his Test debut off the bench last weekend, could make the starting line-up with uncapped 21-year-old Harri O’Connor the only other available tighthead.
South Africa, who won the first Test in Pretoria 32-29 on 2 July, name their name on Tuesday with many stars like captain Siya Kolisi and Cheslin Kolbe who were rested last weekend, poised to return.