Hodgson, 67, guided England to 2-1 defeats by Italy and Uruguay and a 0-0 draw with Costa Rica on their way to early elimination from the tournament.
"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."
(Reporting by Tim Collings, editing by Ed Osmond)
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