Shaw was the victim of an ugly two-footed challenge from Hector Moreno in the 15th minute of United's Champions League group stage loss in Holland.
The 20-year-old was left in agony on the turf at the Philips Stadion following the tackle - which went unpunished - and he was carried off the pitch on a stretcher while receiving oxygen.
Louis van Gaal looked visibly upset when he tapped the England defender on the shoulder as he was taken away down the tunnel to hospital.
And after the match, the United manager was clearly shaken by the incident.
"He has a double leg fracture," Van Gaal said
"He is in hospital
He will be transferred (to the UK) tomorrow (Wednesday) with us or maybe a day later and then operated on in Manchester
"It is awful
He was crying."
Van Gaal was clearly unhappy with the challenge, but refused to condemn the Mexican defender, who played for him at AZ.
"If I say it is a penalty and a red card I am a bad loser, every word is interpreted in the wrong way," he said.
"But it was a very bad tackle with two legs."
Shaw took to Twitter to thank those who had wished him a speedy recovery.
Shaw said: "Thank you everyone for your messages , words can't describe how gutted I am, my road to recovery starts now, I will come back stronger."
At first sight, it seemed Shaw's season - and his hopes of going to Euro 2016 - was over.
But Van Gaal still harbours hope the England international will play again this season.
"I am not a doctor so I cannot say that, but when you have a double fracture it is I think until six months," Van Gaal added.
"It is about how it develops
You can never say that in advance.
"He will not play in the group phase I think
I hope he can play again this season."
If Shaw needs any encouragement he can look to his United team-mate Antonio Valencia, who was back in action five months after suffering a broken leg and dislocated ankle against Rangers in the Champions League in 2010.
United took the lead on Tuesday thanks to an exquisite solo goal from PSV old boy Memphis Depay, but the hosts hit back through Moreno, who headed in a corner after losing Matteo Darmian in the box.
After the restart, Luciano Narsingh headed in after drifting away from Shaw's replacement Marcos Rojo in the box.
Juan Mata, Depay and surprise starter Anthony Martial were all guilty of spurning opportunities during the Group B match in Holland.
"We can only blame ourselves because we have created so many chances," Van Gaal said.
"Today we could have scored much more than that but PSV were very effective because they scored out of nothing."
Source : PA
Source: PA