Luis Suarez opened the scoring in the first half but England were on course for a draw after Wayne Rooney equalised from close range late in the game.
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“No. 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 tried to,” said Hodgson.
“I'm bitterly disappointed, of course, but I don't feel I need to resign, no. 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.”
England now need to beat Costa Rica in their final group game to stand any chance of reaching the knockout phase, and will only then climb out of Group D if Italy win their remaining two games.
"Our chances are unbelievably slim," admitted the 66-year-old.
"But to be sure of continuing we needed a result today – a draw or a victory – and we didn't get it.
"I thought we controlled Suarez well in general play. He did very, very well to get away to the back post for the first goal, but I thought for long periods of the game we kept him very quiet.
"The second goal, as you probably saw, was an unfortunate flick off Steven Gerrard’s head leaving him through with the goalkeeper and he’s always going to score from that area.
"Ideally, your only plan would be to not allow the flick to put him through with the goalkeeper.
"Two chances came his way and being the top player he is he took both chances and that’s probably ended our chances of staying in the competition.
"I think in both the games we’ve shown some elements of playing some good football but results decide everything and the results have gone against us.
"We're more than disappointed, we're devastated.
"We believed we could do enough to get a result in this game and having worked so hard to counter that first goal in the first half, to get back to 1-1, we fully believed we'd go on to win or at least draw."