Puyol dislocated his left shoulder towards the end of the Nou Camp club's 2-0 win over Benfica in Portugal on Tuesday night.
The 34-year-old was stretchered off and then flown home where he had tests carried out and spent a night in hospital.
And Barca have confirmed the Spanish defender will not be available for the Group G double-header against the Scottish champions.
An official club statement said: "The club captain will undergo conservative treatment and is expected to be out for about eight weeks."
With Barca facing Real Madrid in 'El Clasico' this weekend, as well as Celtic in back-to-back European matches, the news is a blow to coach Tito Vilanova. He is already without Gerard Pique.
Vilanova stated: "We've always had problems with our central defenders, but we'll move forward.
"It's not just that we've lost Puyol for the Clasico on Sunday, but we've lost him for other matches as well.
"The injuries mean we will have to play players in positions that aren't their own, but we'll use our youth players. That's why we have them."
Puyol vowed to return quickly on his Twitter account. He said: "I want to thank everyone who looked after me. I'll soon be back to my best."