"There have been plenty of dark moments, times where looking back you don't believe fate has put you in that situation," he told Sky Sports News.
"The fact is I would quite happily relinquish this job any time my employers think they don't want me anymore, or when I feel the players don't have the respect necessary or feel they don't want to continue playing for me.
"In that moment, if it's time for me to walk away, I will do so, but this time it was made clear to me that wasn't the situation (and) that the FA wanted me to continue in the job.
"The feedback from the people around the team was the same. So, therefore, there was never any doubt in my mind that I shouldn't continue.
"You only walk away when, A, you can't stand the pressure or, B, that you don't feel that you are good enough to do the job and, in that situation, I can safely and strongly say that I never felt that at all."
Hodgson said England's preparations were good, but conceded he faced a major rebuilding task for Euro 2016.
"We have a lot of rebuilding to do," he said. "Not only of our team but also rebuilding all the faith that people showed in us."
The tournament performance of one point from three matches was the worst ever by the 1966 winning nation. It was the first time they had lost two group matches since 1950, the first time they had not won a match and been eliminated at the group stage since 1958.
"I don't believe in the nonsense spoken about the identity of certain countries and how they play," he said. "I don't see the vast amount of difference between how we try and the way Germany try to play.
"Now there is a major difference between the 11 players in the German shirt and the 11 players in the England shirt, but if we are talking about how they attack or how they keep possession and seek for openings when they have got the ball, I don't see a vast amount of difference in philosophy or style."
He said football was not "an exact science" adding that "random things that happen in the game make all the difference."
Hodgson also revealed that he had identified the next England captain in his own mind, but had not yet told him.
"I have half chosen him," he explained. "The person who is captain for the Norway game (Sept 3, friendly) will be the person I will be designating to hopefully be the captain in 2016."
He suggested he was not turning to Frank Lampard, 36, who was England's vice-captain in Brazil. Having left Chelsea, Lampard has started a six-month loan period with Manchester City before joining New York City.
"It's not easy, when you're playing in America, to go back and forward for the odd England game," said Hodgson. "It's not to say it's not a possibility but realistically we know that Frank - we've had this discussion - will be 38 in 2016.
"We can't consistently look backwards on these very, very great players who have played over 100 times for England and expect it to last forever."
England begin their Euro 2016 qualifiers with a trip to Switzerland on Monday, Sept. 8. They will also face San Marino, Estonia, Lithuania and Slovenia in Group E as they seek to qualify for the finals in France.
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