Another defeat against Costa Rica on Tuesday would increase public pressure on Hodgson, but the Football Association says there is no chance of the England manager losing his job regardless of what happens in Belo Horizonte.
Hodgson's tactics and his team selection have been criticised by former internationals including Paul Scholes and Gary Lineker.
But Hodgson has complete faith in his abilities.
"There's no reason for me to have any serious doubts about the work I've done over the years and the qualities I bring to the team," said the England manager, speaking for the first time since his team's exit was confirmed.
"I have the backing of the team and the FA and I feel I am the right man to continue."
England suffered one of their worst results in Belo Horizonte 64 years ago when the United States beat Walter Winterbottom's team 1-0.
There would be big public dismay towards Hodgson if England were subjected to another humiliation, but having received the backing of his superiors, the 66-year-old is adamant he will not walk, regardless of what happens on Tuesday.
"I shan't change my mind. I see no reason to resign," Hodgson added.
"I feel allegiance to the players and the staff. I think we work well together.
"I accept this campaign has not been good.
"I'm grateful I've not been made that scapegoat and that people think I can take the team forward, and that's what I will be doing."
England may have made their worst start to a World Cup campaign in history, but there has been no major public backlash towards Hodgson.
There are several reasons for this. Firstly, the list of potential successors is short and unimpressive. Secondly, England have only suffered narrow defeats in Brazil. T hirdly, the way in which England attacked during their defeat to Italy was gripping and exciting.
That is all of little comfort to Hodgson, who admits England's exit was the lowest point of his 39-year long management career.
Hodgson said: "I had two European final defeats. That was pretty tough. And losing the job at Liverpool was as well.
"But this is England. A job that means so much to me and so many people, a massive job, so yes this is the worst.
"I don't think anyone has tried to shirk away from our responsibility. We played two good teams. We believed we were good enough to beat them, but we weren't. So there's a lot for us to do, a gap to make up."
Hodgson has had to put his pain to the back of his mind over the last few days as he is determined not to end the tournament with three straight defeats.
"I certainly haven't been moping around staring at the wall," Hodgson said.
"I had a bad night and a bad day the following day. My job leading the team is to pull myself out of that and make sure the players aren't suffering in the same way.
"It is to give them some of the qualities I've got. So I've been working hard to make sure people do get out of it, try and find the mental, physical and emotional strength to move on.
"A period of grieving is necessary. In three or four weeks' time, we may be miserable again, but that's part of our lives.
"But we can't chase the past. We must live in the present. What we can change is what we do in the future."
The immediate future entails a match against Costa Rica on Tuesday - and Hodgson has confirmed he will rotate his team for the final Group D game.
Gerrard will start on the bench, meaning Frank Lampard will captain the side in what could possibly be his final England game.
Ross Barkley and Luke Shaw will start the game. Leighton Baines will miss out with a groin problem and Alex Oxlade-Chamberlain has suffered a recurrence of his knee injury and will therefore play no part in Belo Horizonte.
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