The result means that England's chances of qualifying for the knockout stages are now out of their hands, but Hodgson insists that he still wants to manage the country.
"I don't have any intention to resign," he is quoted as saying by FIFA.com. "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. I've been really happy with the way the players have responded to the work we've tried."
England's final group match is against Costa Rica in Belo Horizonte next Tuesday.