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Shaw, 20, will have an operation on his return to Manchester, the club said.
He received oxygen while being treated on the pitch for 10 minutes after Hector Moreno's tackle.
"It is awful. It was a very bad tackle. He had the oxygen mask on and was crying in the dressing room," said manager Louis van Gaal.
Mexico defender Moreno sent his "strength" to the left-back and wished him a "fast recovery".
The 27-year-old, who broke his leg during last year's World Cup, added: "I've been through this before and I know how it feels."
Shaw moved to Old Trafford from Southampton in June last year in a £31m deal and made 20 appearances in an injury-interrupted first season.
He has started all of United's eight games this season, as well as playing in England's European Championship qualifiers against San Marino and Switzerland.
Yet with just nine months to go until Euro 2016, Shaw has suffered a broken tibia and fibula.
When Arsenal midfielder Aaron Ramsey suffered a similar injury against Stoke in 2010, he was ruled out for nine months. The Welshman tweeted: "Fingers crossed for Luke Shaw. Get well soon."
Less than an hour after suffering the injury, Shaw wrote on Twitter: "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."
Former PSV player Memphis Depay scored the opener for the visitors in Eindhoven, before Moreno headed in an equaliser on the stroke of half-time.
Luciano Narsingh then nodded home in the second half to secure victory.
Mexican centre-back Moreno escaped censure for the challenge on Shaw, and Van Gaal was unhappy referee Nicola Rizzoli took no action.
"A boy who came to Manchester United at 18 years old, it was very difficult, and the next season he plays fantastic and then that happens like this," said the Dutchman.
"When I say it is a red card and penalty, you will say I am a bad loser. Every word I say is taken in the wrong way. It was in the 18-yard box and it was very bad tackle with two legs.
"I'm not a doctor so I cannot say, but when you have a double fracture it is four to six months. But you can never say that in advance.
"He won't play in the group stage, I don't think. I hope he will play again this season."
PSV manager Phillip Cocu expressed sympathy for Shaw, and said Moreno had not intended to hurt him.
The former Netherlands midfielder said: "We are all sorry about Shaw's injury.
"Hector is a player who went for the ball and was not out to injure anyone but it didn't look good."
Analysis from chief football writer Phil McNulty: The serious injury sustained by Luke Shaw is not only a devastating blow to the 20-year-old, who had made such an outstanding start to this season, but also to Manchester United and manager Louis van Gaal.
Before the start of the campaign Van Gaal said: "I think that this will be the season of Luke Shaw - that I believe." Now those hopes have been cut down by an injury that looked so bad that the defender will be sidelined for months.
In a somewhat stodgy start to the new season for United, Shaw had shone as he looked leaner, sharper and more settled than during a mixed first year at Old Trafford.
The sight of the left-back being taking on off an stretcher also capped a poor day for England manager Roy Hodgson after Arsenal midfielder Jack Wilshere was ruled out for three months as he requires surgery on a hairline fracture of his left leg.
Shaw was inundated with messages of support on social media.
Goalkeeper David De Gea said "defeat hurts, but Luke Shaw's injury hurts even more".
Defender Chris Smalling said his "thoughts and prayers" were with his team-mate and midfielder Ander Herrera wants Shaw to "come back soon, my friend".
Full-back Luke Shaw suffered a double fracture of his right leg in Manchester United's 2-1 Champions League loss at PSV Eindhoven.