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Mr Erdogan accused the moderator of not allowing him to speak and said he did not think he would return to Davos.
He was cut off as he attempted to reply to a passionate defence of Israel's actions made by Mr Peres.
Turkey is one of the few Muslim countries to have dealings with Israel, but relations have been under strain since the Islamist-rooted AK Party was elected to power in 2002.
"I do not think I will be coming back to Davos after this because you do not let me speak," Mr Erdogan shouted before marching off the stage in front of Mr Peres, UN Secretary General Ban Ki-moon and an elite audience of ministers and international officials.
He criticised the moderator for cutting him short on the grounds that the debate had gone over time.
"I'm optimistic about Israel. I don’t accept the assumption that Israel is dependent on the global economy and nothing can be done about it. Israel was one of the last countries to enter the crisis, and if we act right, Israel will be one of the first countries to emerge from it."
"What is not reversible is the acquisition of nuclear weapons by a fanatic radical regime ... We have never had, since the dawn of the nuclear age, nuclear weapons in the hands of such a fanatical regime," he said.