The son of former President George Senior and brother of George Junior is a Catholic but still took aim at the pontiff as he prepares to issue an encyclical on the environment.
"I hope I'm not going to get castigated for saying this by my priest back home," he said, "but I don't get my economic policy from my bishops or my cardinals or my pope."
Bush, who converted to Catholicism 25 years ago was speaking at a town hall event in New Hampshire after he announced his intention to stand for the 2016 presidential election.
Pope Francis is set to publish an encyclical on climate change on Thursday although the letter has been partly leaked already.
But Bush said religion "ought to be about making us better as people and less about things that end up getting into the political realm."
He has previously said the evidence on climate change was not complete.
He's not the first Republican presidential hopeful to criticise the Pope.
Earlier in June Rick Santorum, also a Catholic, said the Holy Father should leave "science to the scientists".
"When we get involved with controversial and scientific theories, I think the Church is not as forceful and not as credible," he said.
"I've said this to the Catholic bishops many times when they get involved in agriculture policy, or things like that, that are really outside of the scope of what the Church's main message is, that we're better off sticking to the things that are really the core teachings of the Church as opposed to getting involved in every other kind of issue that happens to be popular at the time."