The 68-year-old had been in charge for four years after replacing Italian Fabio Capello but has won just three of 11 games in major tournament finals.
Iceland - with a population of just 330,000 - were among the lowest-ranked teams in France at 34 in the world.
"I'm sorry it will have to end this way but these things happen," Hodgson said.
"I hope you will still be able to see an England team in a final of a major tournament soon.
"Now is the time for someone else to oversee the progress of a hungry and extremely talented group of players. They have done fantastically, and done everything asked of them."
FA chairman Greg Dyke had stated he would only stay on if England "do well" in France, indicating that meant at least reaching the quarter-finals - but they needed to beat Iceland to do that.
In a statement issued following Hodgson's resignation, the FA said: "Like the nation, we are disappointed to lose this evening and that our run in Euro 2016 has come to a premature end.
"We back Roy Hodgson's decision to step down as England manager and will discuss next steps imminently."
England qualified for Euro 2016 with a 100% record in their group. They started the finals with a 1-1 draw against Russia before beating Wales 2-1 and drawing 0-0 with Slovakia in their final Group B game - results that left them in second place and in a tougher half of the draw.
Hodgson faced heavy criticism following the Slovakia game for making six changes to the side that had beaten Wales.
Under Hodgson's guidance, England had reached the quarter-finals of the last European Championship, losing to Italy on penalties, before a dismal showing at the 2014 World Cup where they went out at the group stages without winning a game.
Speaking to reporters following Monday's last-16 defeat in Nice, Hodgson confirmed his assistants Ray Lewington and Gary Neville would also be leaving their posts.