Mr. Bush, addressing more than 160 presidents, prime ministers and kings gathered for three days of U.N. General Assembly meetings, was seeking to sell his blueprints for spreading democracy in Iraq and elsewhere, overhauling the United Nations and expanding trade.
"The terrorists must know that wherever they go they cannot escape justice," Mr. Bush said to world leaders who sat silently throughout his speech.
CBS News White House Correspondent Mark Knoller reports Mr. Bush began his speech by offering America's thanks to the more than 115 nations that have responded with offers of aid to the U.S. following Hurricane Katrina.
"Your response, like the response to last year's tsunami, has shown world more compassionate and hopeful when we act together," he said.
There is broad opposition to the U.S.-led war in Iraq among the world leaders gathered in New York to help the U.N. celebrate its 60th anniversary; many also would rather hear Mr. Bush finally relent and support an international treaty on global warming or promise to donate foreign aid at a level more proportionate to other rich nations.
However, the president's focus was again on the effort to defeat terrorists and the importance of supporting Iraq.
Mr. Bush pressed for Security Council approval of a resolution calling upon all nations to take steps to end the incitement of terrorist acts and asked nations to agree to prosecute and to extradite anyone seeking radioactive materials or nuclear devices.
"We must send a clear message to the rulers of outlaw regimes that sponsor terror and pursue weapons of mass murder: You will not be allowed to threaten the peace and stability of the world," Mr. Bush said. "Confronting our enemies is essential, and so civilized nations will continue to take the fight to the terrorists."
Mr. Bush urged the elimination of agricultural tariffs and other barriers that he said distort trade and stunt development. The goal, he said, is to open markets for farmers around the world.
"Today I broaden the challenge by making this pledge: the United States is ready to eliminate tariffs, subsidies and other barriers to free flow of goods and services as other nations do the same," he said. "It's the key to overcoming poverty in the world's poorest nations. It's essential we promote prosperity and opportunity for all nations. By expanding trade we spread hope and opportunity to the corners of the world and we strike a blow against the terrorists who feed on anger and resentment."