England returned home on Wednesday after finishing bottom of Group D with a single point following 2-1 defeats by Italy and Uruguay and a 0-0 draw with Costa Rica on Tuesday.
Italy coach Cesare Prandelli resigned after his side were beaten to a last-16 berth by Uruguay but Hodgson said he would not follow suit.
"He's been there a long time," Hodgson said of Prandelli's decision to step down during a tetchy post-match briefing with journalists at Belo Horizonte's Estadio Mineirao.
He reaffirmed that the English Football Association has asked him to continue. "I'm very happy that they want me to continue," Hodgson added.
"I don't intend to walk away from things. I've stood up to the criticism, I've stood up to the comments,” he said.
"That's about as much as I can do and I don't want to be compared with other people. The reason I'm staying on is because I'm not a quitter,” he further said.
Hodson added: "I believe in this team, I believe that the FA seriously want me to do the job, as do the players, so therefore I'll continue to do it, and I'll try and lead the team to Euro 2016 and I'll try and get good results there."
While Hodgson's youthful side were praised for an enterprising showing against Italy, captain Steven Gerrard felt that they had suffered from a lack of composure in last Thursday's decisive loss to Uruguay.

England's inability to handle pressure has been identified by pundits as a telling factor in the team's downfall but Hodgson says it is unrealistic to expect inexperienced players to demonstrate maturity beyond their years.
"The thing about game management and that type of understanding of the game is that it often comes with age and experience and games," said Hodgson, who described the days since England's elimination as “absolute misery”.
Hodgson also spoke out in defence of Manchester United defender Phil Jones, 22, who produced a wobbly display at right-back against Costa Rica, pockmarked by sloppy technical errors.
"I think he played very well," Hodgson said.
"I thought that both he and (left-back) Luke Shaw played very well in the game, so we've seen different performances there," said the coach.
"I thought he was absolutely excellent and I thought the whole back four, the young back four, played excellently,” he added. 


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