Luis Suarez’s brace condemned England to a second successive 2-1 reverse in Group D following last Saturday’s loss to Italy in their opening World Cup game.
The setback against Uruguay means England are facing up to a group stage exit at the World Cup for the first time since 1958.
To stay in with a chance of reaching the last 16, England are relying on Italy to beat Costa Rica on Friday, and then Uruguay on Tuesday.
England will also need to beat Costa Rica in Belo Horizonte and hope there is a big enough swing in goal difference to send Hodgson’s side into the knockout rounds.
Hodgson, whose contract runs until the end of Euro 2016, does not feel he needs to quit if England’s time in Brazil comes to an end at the first hurdle.
"I don't have any intention to resign," said Hodgson.
"I'm bitterly disappointed, of course, but I don't feel I need to resign, no."
Hodgson, who looked absolutely distraught after the match, admits that whether he stays or goes is out of his hands, though.
He continued: "On the other hand, and if the FA think I'm not the right man to do the job, that will be their decision, and not mine.
"I've been really happy with the way the players have responded to the work we've tried to do."
Hodgson was upbeat about England's chances in the build-up to the World Cup, and there were flashes of brilliance in their play against Italy on Saturday.
But there was barely any inkling of optimism in the 66-year-old's voice after the match in Sao Paulo.
"Our chances (of progressing) are unbelievably slim," he said.
"It will depend on Italy winning their next two matches by a good number of goals, and us beating Costa Rica by the requisite number of goals.
"To be in with a chance of continuing we really needed a result today, a draw or a victory, and we didn't get it."
Asked if he thought England were outplayed, Hodgson was blunt in his response: "No."
The England manager gave a more expanded answer to a question on Suarez's efforts.
The controversial Liverpool striker had knee surgery nearly five weeks ago, but he managed to beat Joe Hart twice and chased England's back four all night.
Hodgson believes his team did a decent job keeping Suarez quiet for most of the evening, saying: "I thought we controlled Suarez well in general play.
"He did very well to get away to the back post for the first goal but we kept him very quiet for long periods of the game.
"We're normally used to seeing him a lot more active in and around the box. The second one was an unfortunate flick from Steven Gerrard to put him free for the goalkeeper.
"He doesn't miss from them. To make certain, you don't allow him the flick to put him through on the goalkeeper."
Suarez's place in the World Cup was in doubt when he suffered a knee injury in a training camp, but Hodgson always expected him to line up in Sao Paulo.
"We always heard that he would probably be fit for some part of the World Cup, and was targeting at least the second game," the former Liverpool manager said.
"What the Uruguayans wanted from him today, they got. He had two goal chances and he took them both."
England flew back to their base in Rio de Janeiro immediately after the match. The party will fly to Belo Horizonte on Monday before their final game 24 hours later.
England are now hoping for a huge favour from the Italians but Gerrard admits his side's hopes of staying in Brazil are slim.
"Italy are a fantastic team and capable of winning both games but we're clutching at straws," the Three Lions captain said.
"The position we're in is the position we didn't want to be in coming into the tournament, relying on other people's results. It's very difficult at the moment.
"We'll have to wait and see what happens in tomorrow's fixture. The manager will have to wait and see what happens before deciding what he is going to do.
"If there's nothing to play for then I don't know if he'll play a completely different team. But it's a very tough moment, at this moment."