Cheney, who arrived in Japan on Tuesday, was given a 19-gun salute as he boarded the USS Kitty Hawk at this U.S. Navy base just south of Tokyo.
"The American people will not support a policy of retreat," Cheney told about 4,000 troops in the hangar bay. "We want to complete the mission, we want to get it done right, and then we want to come home, with honor."
Cheney said his brief visit to Japan was a gesture of appreciation for Tokyo, which has been one of Washington's most valuable allies in the war on terror by offering non-combat troops to assist U.S. efforts in both Iraq and Afghanistan.
Japan pulled its troops out of Iraq last year. The country still operates airlifts in the region in support of the U.S.-led forces, a mission set to end in July. Japan's navy has also provided fuel for coalition warships in the Afghan war since 2001.
Cheney also commented on's announcement that Britain would begin withdrawing its troops from southern Iraq, calling it a sign of progress.
"Well, I look at it and see it is actually an affirmation that there are parts of Iraq where things are going pretty well," Cheney said in an interview with ABC.
Cheney said he had spoken with a friend who traveled recently from Baghdad to Basra and "found the situation dramatically improved from a year or so ago."
In Japan, Cheney met with Emperor Akihito and Empress Michiko and Foreign Minister Taro Aso before holding talks with Prime Minister Shinzo Abe.
Cheney and Abe discussed their countries' stances on North Korea, which has rattled the region with its nuclear ambitions but earlier this month agreed to shut down its main nuclear reactor in exchange for energy aid and other incentives.
"We are cooperating to protect our countries from the dangers of nuclear proliferation," Cheney told Abe at a photo session prior to their talks. Earlier, Cheney said the accord with North Korea was "a correct first step," but pledged to keep up pressure on the communist regime to dismantle its nuclear program.
Despite the breakthrough, Japan has declined to contribute aid to the North until the issue of past abductions of its citizens is resolved. North Korea has admitted kidnapping Japanese in the late 1970s and '80s, but Japan believes it has not provided a full accounting of the victims.
Cheney told Abe at the photo session that the U.S. respects Japan's position and wants "to seek a resolution of the tragic case of Japanese abductees." He was to meet the parents of one of the abductees on Thursday before flying to Guam and then on to Australia, his final stop.
Cheney and Abe also discussed China's anti-satellite weapon test last month. They agreed it demonstrated the country's lack of transparency over its military and that Beijing needed to become a more "responsible and constructive force in international society," a Foreign Ministry official said later at a briefing on customary condition of anonymity.
Cheney relayed an invitation from President Bush for Abe to visit the U.S., the Foreign Ministry official said.
Also Wednesday, Cheney and Tokyo's top government spokesman, Yasuhisa Shiozaki, reiterated their countries' intentions to collaborate closely on missile defense and the realignment of U.S. troops stationed in Japan.
The U.S. has about 50,000 troops stationed in Japan under a mutual security pact that dates back to the 1960s. Tokyo and Washington have been reworking that alliance to make the presence more effective. They are also developing a joint missile defense shield.