Shaw was substituted midway through the second half which left manager Ole Gunnar Solskjaer furious with the Premier League over the scheduling of the lunchtime kick-off at Goodison Park.
United travelled to Merseyside less than 72 hours after arriving back from Turkey for a Champions League group stage defeat to Istanbul Basaksehir.
Solskjaer accused the Premier League of ‘setting United up to fail’ with the quick turnaround in games and will now be without Shaw until the middle of next month.
The 25-year-old is likely to miss upcoming Premier League games against West Bromwich Albion, Southampton and West Ham, as well as as Champions League clashes with Istanbul Basaksehir, Paris Saint-Germain and RB Leipzig.
Solskjaer must now turn to either new signing Alex Telles, who recently recovered from coronavirus, or youngster Brandon Williams to fill in at left back.
“Our boys were set up to fail today with the scheduling,” Solskjaer said after the final whistle at Goodison Park. “How can you expect players to perform with the 12.30 kick off time? It’s an absolute joke.
“Of course we tried to move this game. What sense was there playing Saturday lunchtime when we could play Sunday because of the international break afterwards?
“I pushed my club to really fight on this one to give us half a chance. But it was just quashed and we were told: ‘No, it’s been selected’.”
Marcus Rashford and Victor Lindelof suffered minor injuries in the victory over Everton and will join up with their respective national teams late as a result.