The Red Rose crashed out of the tournament yesterday with a 19-12 defeat to France - a disappointing result having reached the last two finals and winning the tournament in 2003.
Rob Andrew, operations director at the RFU, will now undertake a comprehensive "root and branch" review of England's failed campaign before deciding on the future of coach Martin Johnson.
Criticism has already been levelled at the England team and management following a number of disappointing performances both on and off the field.
However, Thomas, who has already survived two votes of no confidence, is refusing to consider stepping down.
"I'm not in the business on turning my back on conflict," he told BBC Radio 5 Live's Sportsweek.
"I've never done it in life and I'm not going to start doing it later in life.
"I am not going to resign over what's gone on because I don't believe I have anything to resign over. Other people may judge it differently and they are entitled to do so."
And asked what his position would be is there was another vote of no confidence, he said: "I'm very disappointed that anyone that has England rugby at heart as opposed to a personal agenda, whatever they think of me, would even contemplate trying to put that into the public domain while we're in the middle of a Rugby World Cup campaign.
"If they want to start it now the campaign is over, so be it.
"I've survived two votes of no confidence at the last two council meetings and that's all I can do really.
"If they want to pursue it, so be it. My heart is in always in doing what I believe are in the best interests for England and supporting the England team.
"It's a matter for others to judge if that's what they want and if they want someone else to do it."
Thomas also refused to call for Andrew and Johnson to fall on their swords.
Johnson's current contract expires soon and the former captain is set to consider his future himself, while Andrew is expected to assume another new role within the RFU shortly.
"I can't say whether Rob would leave. We've got to do a review and if there are failings in the rugby departments we need to identify those failings and determine on the course of action that will correct the position," Thomas added.
"As far as Martin's concerned, we'll get the review done. I'm not going to make it (the decision) for him.
"He needs some time to think about it and we all need some time to think about it.
"We delivered in the Six Nations, we started this campaign well, we slipped against Wales and we have blatantly failed to deliver in this World Cup.
"I wouldn't agree that he's likely to continue or not continue. I honestly couldn't call that. It's a matter for him to make that decision as to whether he wants to continue into 2015.
"He's big enough to look after himself in that respect. He has my backing, certainly."
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