Hodgson's contract was due to expire at the conclusion of the finals and the shock 2-1 defeat in the last 16 in Nice prompted the 68-year-old to announce his departure.
"I'm going to read a statement, so I will not be misquoted," he said in the post-match media conference, before a fuller version was published by the Football Association (FA).
"I'm extremely disappointed. We haven't progressed as far as I thought we were capable of and that's obviously not acceptable."I'm proud of the work my coaching staff and I have achieved during our time at the helm with England. The transition of the squad whose average age is 30 to now being the youngest in the tournament is both remarkable and exciting for the future of English football.
"I would have loved to have stayed on for another two years, however I am pragmatic and know that we are in the results business.
"My contract was up after the Euros. So now is the time for someone else to oversee the progress of this young, hungry and extremely talented group of players.
"They have been fantastic and they have done everything that has been asked of them.
"When I arrived I was told that players didn’t turn up to play for their country, or that they pull out at the last minute but I’ve not seen any of that. These players love to play for their country and their commitment has been unquestionable.
"Ray Lewington and Gary Neville have asked me to speak on their behalf. They arrived with me as part of my coaching team and they’ll leave with me. I'd like to thank them for their dedicated support and for the major part they've played in our team preparation.
"I'd like to thank all the support staff, players, The FA and of course the fans – it’s been a fantastic journey these four years and it’s one I’ll look back on with pride.
"Finally I'd like to thank the media for the support that you’ve given me during those four years.
"I’m sorry it’s had to end this way with another exit from a tournament but these things happen and all I can do is wish everybody all the very best and hope that you will still be able to see an England team in a final of a major tournament fairly soon…we’ve been unable to do it. Thank you very much."
After reading the statement, Hodgson immediately left the news conference and there were no further questions.
Shortly after Hodgson's resignation, the FA published the following statement: "Like the nation, we are disappointed to lose this evening and that our run in Euro 2016 has come to a premature end.
"We had high hopes of progressing through to the latter stages of the competition and accept that we have not met our own expectations or those of the country.
"We back Roy Hodgson's decision to step down as England manager and will discuss next steps imminently.
"For now, we congratulate Iceland for reaching the quarter-final and wish them well against France at the weekend.
"Finally, our sincere thanks go to the travelling fans for their incredible support tonight and to everyone back home for getting behind the team."
Things began brightly for England with Wayne Rooney's fourth-minute penalty, but goals from Ragnar Sigurdsson and Kolbeinn Sigthorsson hastened the exits of both the team and then the manager.
Gareth Southgate, manager of England's Under-21 side, is the early front-runner to replace Hodgson, with Eddie Howe and Alan Pardew also among the contenders, according to the bookmakers.