Speaking in Singapore, Mr Bush said the US would resist "old temptations of isolationism and protectionism".
Despite opposition in the Democrat-held Congress, he defended open markets, saying an Asia-Pacific free trade area must be "seriously considered".
He also urged North Korea to seek "a peaceful path" in the nuclear crisis.
A large part of Mr Bush's speech on Thursday at the National University of Singapore was devoted to the opportunities afforded by global free trade.
| || America believes in free and fair trade and we will continue to open up new avenues to commerce and investment across this region |
George W Bush
"We hear voices calling for us to retreat from the world and close our doors to these opportunities," he said.
"These are the old temptations of isolationism and protectionism and America must reject them. We must maintain our presence in the Pacific. We must seize on our common opportunities."
Mr Bush vowed that the US would "remain engaged in Asia because our interests depend on the expansion of freedom and opportunity in this region".
Ahead of his three-nation Asian tour, Mr Bush had suffered a setback when an historic bill to normalise trade with communist Vietnam failed in the House of Representatives.
The BBC's Jonathan Head in Singapore says Mr Bush's Asian hosts are aware that in the last two years of his presidency, his hands may be tied by a Democrat-controlled Congress with more protectionist instincts.
Nevertheless, Mr Bush defended his trade beliefs.
He said: "Recently some [Asia-Pacific Economic Cooperation] members have advanced the idea of a free trade agreement for the entire Apec region. I believe this idea deserves serious consideration."
Mr Bush's tour will take in this weekend's 21-nation Apec summit in Vietnam. He will also visit Indonesia and next week travel to Europe for a Nato summit.
On North Korea, Mr Bush called on Pyongyang to take "concrete steps" to help resume high-level talks aimed at curbing its nuclear programme.
He urged regional countries to make it clear that any North Korean proliferation of nuclear technology to terrorists or hostile regimes "would not be tolerated".
Mr Bush also called for co-operation in finding new affordable and reliable supplies of energy. "It is in the world's interest to work together to end our addiction to oil," he said.
After Mr Bush arrived in Singapore on Thursday, he and the First Lady visited Singapore's Asian Civilisations Museum and were treated to traditional Javanese and Singaporean music.
Mr Bush briefly tried to play a saron - an Asian-style xylophone - but said: "I'm going to quit while I'm ahead."
The president is likely to hold a number of meetings on the sidelines of the Apec summit in Hanoi at the weekend.
He will meet Japan's new Prime Minister Shinzo Abe, South Korean President Roh Moo-hyun, Russian leader Vladimir Putin and Chinese President Hu Jintao.