The 30-year-old is set to be sidelined for between four and five months after undergoing successful surgery on Monday on the injury he sustained during the first half of Thursday's Club World Cup semi-final victory over Al-Sadd in Japan.
A Barca statement on Monday night read: "After surgery on his tibia, David Villa thanked friends and fans for the support shown to him these past couple of days. Among those he thanked was [assistant coach] Tito Vilanova, who visited the striker at hospital."
Spain's all-time record scorer said on www.fcbarcelona.cat: "My head has been recovering for two hours. I'll start to rehabilitate this leg soon.
"I'm really looking forward to getting well so I can return and fight on the pitch.
"With a lot of work and with people's help, you'll see me on the football field soon, enjoying myself like I did before."
Villa faces a race to be fit in time for next summer's European Championships.
According to Dr Ramon Cugat, who was one of the doctors who operated on Villa on Monday afternoon, the Asturian is unlikely to feature again during Barca's Primera Division and Champions League campaigns but hopes the forward will be fit for Euro 2012.
Cugat said: "The hope of him (Villa) and everyone will be that he can play at the European Championships.
"The chances that Villa will play this season are small but you can't rule out anything."
The end of the Primera Division season is scheduled for the weekend of May 12, with the Champions League final taking place in Munich on May 19, while Euro 2012 begins on June 8.
"The final in Munich is a little tight. I think he will be out for four or five months, but we have to wait and see how he progresses," Cugat added.
"If I set a (return) date, I would surely be wrong. He is keen to be able to play in the both the Champions League final and the European Championships. He wants it all.
"In four or five months we hope to have him ready for the European Championships."