Luis Suarez's double for Uruguay condemned England to their second successive defeat in Group D following an opening loss to Italy.
England's future hangs by a thread with Italy having to beat both Cost Rica and Uraguay and England beat Costa Rica as well by enough goals to get through on goal difference.
But Hodgson, although hugely disappointed said afterwards:
"I don't have any intention to resign.
'I've been really happy with the way the players have responded to the work we've done.
"Our chances of going through are unbelievably slim," he went on..
"We depend on Italy winning their next two matches and us beating Costa Rica by the requisite number of goals."
QPR boss Harry Redknapp, who was strongly tipped to take over from Capello at one stage, said Hodgson should remain in charge.
"Roy couldn't have done any more," he told Radio 4's Today programme. "He's prepared the team well, worked hard, you can see they've got a good camp. At the end of the day, we've come up short."
However, Redknapp said he felt the "future of English football" lay with Liverpool manager Brendan Rodgers and Everton boss Roberto Martinez.
"I think there's two young managers on Merseyside who are absolutely fantastic," he said. "When Roy does calls it a day, I'd like to see England play the way Liverpool played this year under Rodgers or the way that Everton have played under Martinez."
Former England striker and BBC pundit Alan Shearer backed Hodgson to take England to Euro 2016.
"Everyone wanted to see the kids given a chance and he went and tried to win the game against Italy," Shearer said. "He should be given the chance to finish what he started."
Graham Taylor, who managed England between 1990 and 1993, also said the FA should "stick with" Hodgson.
"It's a process of building and it takes time," Taylor told Radio 4's Today programme. "Roy has only been in the job for two years and it takes longer than that to put a team together in international football."
England captain Steven Gerrard, who was instrumental in allowing Suarez's second strike insisted Hodgson was a "fantastic manager".
Gerrard added: "Italy are a fantastic team and capable of winning both of their remaining games, but we're clutching at straws. We need to be professional and pray for that scoreline to go for us."
One glimmer of hope for England is that with both Costa Rica and Uraguay having won a game each defeat to either could see Italy lose out on goal difference if all sides finished on six points.