Speaking in Los Angeles, where he is on a tour of the American Episcopal Church, Dr Carey said: "Whether the new millennium will be one of peace or war will depend to a large measure on the ability of the great religions, and Christianity in particular, to draw from within themselves all which makes for peace.
"If religions are not dying out, and may be on the increase in many parts of our world, the religious leadership has a responsibility to resist anything that is done in the name of religion which denies the true ends of religion. I think of extremism which ends in murder, and violence.
"Sometimes, when acts are perpetrated by fundamentalists, I am saddened that few leaders of such faith communities condemn the atrocities. People should not hide behind religious beliefs to justify acts of terrorism."
His speech will be seen as containing criticism of some Muslim countries, especially those which deny to Christians the liberty of worship they demand for their own subjects abroad. "Muslims, Sikhs, Hindus and others have equal rights to worship freely in the West," he said. "However, this must apply equally to the rights that Christians should have in places where they are a minority."
Professor Akbar Ahmed, of Selwyn College Cambridge, who last week preached the first Muslim sermon in a Cambridge College Chapel, said: "Dr Carey is trying to do the right thing, but underneath, I suspect he is seeing this relationship in terms of . . . confrontation when it should be dialogue. Muslims see people like the Archbishop as still harbouring some of the agenda of the crusade."
As if to illustrate this, Professor Ahmed has been denounced this week by the self-styled Muslim Parliament for preaching in a Christian church. A spokesman for the parliament told an Urdu newspaper that his actions were a preliminary to asking Christian priests to preach in mosques.
Dr Carey has long argued that religion is undervalued in human affairs; and yesterday drew to his aid a controversial Harvard professor, Samuel Huntington, who three years ago argued that the collapse of communism meant that the main rival for the West now is the Islamic world. "The fault lines of civilisations will be the battle lines of the future", Professor Huntington claimed.
This "beguiling hypothesis", said Dr Carey, had been too quickly discounted. Some people thought it exaggerated; others it was politically incorrect. However, he believed that Professor Huntington had grasped "something essential to world peace" when he spoke about the importance of the West understanding the basic religious and philosophical assumptions underlying other civilisations.
Dr Carey argued that the Church of England's difficulties over women priests offered a model in terms of conflict resolution, in as much as opponents of the decision had been accommodated as far as possible, to that the two sides could "live in peace together for the sake of all we have in common".