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.
"I don't intend to walk away from things. I've stood up to the criticism, I've stood up to the comments.
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"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.
"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."