"The operation was done, it went well and it will take many months for him to recover," said Sakari Orava.
He added there was "no chance" of Beckham making the 11 June -11 July showpiece in South Africa for what would have been an English record fourth finals appearance. "That is quite clear," Orava said.
Beckham twisted his left ankle during the final minutes of an AC Milan Serie A game against Chievo at the San Siro on Sunday and limped off the field before being stretchered away in pain and in tears. The 34-year-old midfielder flew to the western Finnish city of Turku in a private plane on Monday and was walking on crutches when he arrived at the sports injury clinic of Mehilaeinen hospital, where Orava operated on him.
Beckham would stay in Finland overnight and could travel to either England or Italy either on Tuesday or Wednesday, Orava said. Earlier England manager Fabio Capello commented: "I spoke with him after the game on Sunday night to offer my support. "David is a great professional and has worked very hard to be ready for the World Cup, so missing it will be a big blow."
Capello later tried to give Beckham some encouragement by saying the player was in his thoughts for England's UEFA EURO 2012 qualifying campaign, which is due to start in September. "I hope he will be fit for the Euros because he is always one of the best players," said Capello.
Beckham speaking before Monday's operation said he hoped to make a swift and full recovery. "I am on my way to Finland to see a specialist and have a scan on the injury," he said. "I am upset but want to thank everyone for their messages of support. I hope to make a swift and full recovery."
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AC Milan vice-president Adriano Galliani said that the player will be out for five or six months. "I saw him and he was suffering," Galliani added. "In the dressing room, I took him in my arms and told him that if he wanted, he will be with us next year."
Beckham had admitted earlier on Sunday he was far from certain to win a place in England's squad for the 2010 finals. He was an unused substitute in England's recent friendly win over Egypt as Capello opted to look at Theo Walcott and Shaun Wright-Phillips on the right wing.
He joined AC Milan on loan in January in an attempt to convince Capello he could still make an impact at the highest level, but he has been left on the bench several times during his spell at the San Siro. An injury crisis at the club, however, meant he was handed a rare start on Sunday only to suffer the major injury.
Beckham is England's most-capped outfield player with 115 caps. The injury could be the final twist in his football career but Beckham's former Manchester United team-mate Andy Cole said the free-kick specialist would not want to retire in such circumstances. "He'll come back from the injury, I don't think he'll want to retire on this note," former England striker Cole said.