He also noted an Army Stryker brigade from Wainwright is about to deploy to Iraq, saying it will be "heading into a different situation."
"About a year ago people thought Iraq was lost and hopeless.
People were saying let's get out of there, it doesn't matter to our national security. Iraq's changed - a lot," Bush said. "The terrorists are on the run."
And he again asserted that fighting them was not a law enforcement problem. "If it's a law enforcement matter, that means you react after the crime. I think it's important ... to stop the crime from happening in the first place."
"The United States today and tomorrow must stay on the offense and keep the pressure on this enemy and bring 'em to justice so they don't hurt the American citizen again," Bush said.
As Air Force One refueled, the president got an enthusiastic welcome from the military members in a hangar known as the Thunderdome. But for him, there also was a delicate bit of politics at play.
Among those in attendance was Sen. Ted Stevens, whose indictment on corruption charges has roiled Republican politics and prompted some of his colleagues to keep their distance.
Bush acknowledged Stevens briefly and warmly, saying the military has no stronger friend. Josh Bolten, the president's chief of staff, was seen chatting with Stevens after the president spoke.
As a member of Alaska's delegation, Stevens was invited as a matter of protocol.
White House press secretary Dana Perino said Stevens' participation was "absolutely appropriate."
Perino said she did not think Bush had spoken to Stevens since his indictment last week. The 40-year Senate veteran is accused of concealing gifts from a powerful oil services company. He has pleaded not guilty to the seven-count indictment against him.
Stevens faces a tough re-election fight this year.He says he expects to be vindicated before the November general election, thanks to a speedy trial he requested.
Eielson is home to the 354th Fighter Wing, which supports operations in the Pacific.
The president departed after his comments for Seoul, South Korea.
His agenda in Asia this week is front-loaded with trouble on the continent: nuclear worries, political repression, recovery from natural disaster. Then comes plenty of sports.
Bush's last venture as president to the Far East is built around the Olympics in Beijing.
Yet with less than six months left in office, he is also out to show that the United States is engaged in Asia's affairs, and that the economic and security dividends pay off back home.
His enthusiastic plans to attend the Olympics are meant to pay respect to the Chinese people in their moment of glory. Yet as hard as Bush tries to define the games only in the context of sports, there is no escaping the politics of a world event held in a police state.
China, trying to ensure the event is clean of controversy, has only intensified its repression of political dissent, religious expression and press coverage. Bush says he can and will candidly raise concerns about China's human rights record to President Hu Jintao.
Before leaving the White House, Bush ratified a United Nations treaty intended to curb performance-enhancing drug use in sports.
It ensures that the World Anti-Doping Code becomes national law and commits member nations to prevent cross-border trafficking of sporting drugs, support a national drug-testing program and withhold funding from athletes caught cheating.
The UNESCO Convention on Doping in Sport came into force early last year, but has not been ratified by all the countries that pledged to do so. Bush's signature followed Senate approval of the treaty.
"The timing of the United States' ratification, on the eve of the Beijing Olympic Games, is appropriate," Bush said in a statement. "The Convention makes clear that the use of performance enhancing drugs to gain a competitive advantage undercuts the positive attributes of sport."
Given the long travel and time differences, Bush begins his agenda in earnest on Wednesday in Seoul.
The country is a key partner in the six-country coalition striving to rid North Korea of its nuclear weapons. Progress has been stop and start as the world watches to see whether North Korea will come to terms on allowing its nuclear dismantling to be verified.
The timing of Bush's visit to Seoul is a bit better than just a few weeks ago. Public unrest over U.S. beef imports has receded, and the U.S. has reversed course on a decision that angered South Korea regarding some disputed islands between Japan and South Korea.
In Thailand, where a coalition government is enduring rocky times, Bush will spell out his vision for the U.S. presence in the Far East after he leaves office. He will also meet with activists who oppose the repression of the military junta in neighboring Myanmar.
That country, also known as Burma, sustained a cyclone in May that killed roughly 80,000 people and put more than 2 million people in need of aid. Bush will be briefed on recovery efforts during his Thailand visit.
The president caps his trip with four days in Beijing, mixing in a dash of diplomacy with plenty of unstructured time to watch Olympic sporting events. Bush will be joined by members of his family, including his dad, a former president who once served as an envoy to China.
Sixty-three percent in the U.S. said Bush should attend the games' opening ceremonies, according to a CNN-Opinion Research Corp. poll conducted in late July. Yet in the same survey, 51 percent said they think China is a military threat to the U.S., and 70 percent considered China an economic threat.
