Rumsfeld told the troops the war that has cost more than 2,150 U.S. lives was a "test of wills." But he added that previous generations of U.S. soldiers faced similar tests and won.
Over the roar of helicopters, the secretary said most Iraqis supported the fight for democracy.
"In this fight, the vast majority of Iraqis stand on the side of freedom," he said.
In other developments:
The soldier was assigned to the 205th Military Intelligence Brigade and was wounded in an attack while on a routine patrol near the town of Hawijah, the military said. The soldiers name was withheld pending notification of next of kin, and no further details were released.
The video, part of which was aired on the Al-Arabiya satellite channel, showed Mahmoud Suleiman Saidat, a driver for the Jordanian Embassy in Baghdad who was snatched on Tuesday by gunmen.
Saidat read a statement while sitting on the floor, surrounded by three armed men holding automatic weapons and a rocket-propelled grenade launcher. A banner hanging in the background identified Saidat's kidnappers as the previously unknown group, the Hawk Brigades.
As U.S. military forces begin to scale back in Iraq, their growing focus on training over combat could lead to a decline in casualties just as the American political season begins to heat up.
Rumsfeld's war zone announcement canceling the deployment to Iraq of two brigades is just the beginning of a gradual drawdown of U.S. forces, according to military commanders.
While defense officials would not detail withdrawal plans for the coming year, the latest move drew praise from defense analysts and critics of the administration's war policies as a positive first step in shifting security responsibilities to the Iraqis.
"I think you will see a gradual shifting toward the Iraqis taking more control, and over time the Iraqis will be bearing more of the casualties," said Brookings Institution foreign policy analyst Michael E. O'Hanlon. But he cautioned: "It's not going to be a simple substitution effect. I think we should be braced for it to be quite slow, and 2006 will still be a bloody year in Iraq."
Rumsfeld, who spent the night at the Camp Victory military installation in Baghdad, had breakfast Saturday with a group of American military intelligence collectors and analysts. Rumsfeld has been trying to encourage efforts to tie the U.S. intelligence effort more closely to military and police operations. That meeting was closed to reporters.
Before flying from Baghdad to Balad, north of the Iraqi capital, Rumsfeld was given a classified briefing on the military's efforts to minimize the danger to troops from roadside bombs, which the military calls improvised explosive devices, or IEDs. He also saw a display of IED detection and destruction devices, as well as vehicles with modifications designed to protect soldiers from the explosions.
Lt. Gen. John Vines, the top operational commander in Iraq, showed Rumsfeld a Humvee utility vehicle whose rooftop gunner shield was modified to add ballistic glass that allows the gunner to remain seated without having his field of vision blocked by the mostly metal shield.
In Balad, Rumsfeld was meeting behind closed doors with special operations troops. Balad is a key airfield and the main distribution hub for military supplies.
On Friday evening, he dined privately with several leading Iraqi politicians, including Ahmad Chalabi and others who aspire to be the country's next prime minister.
Gen. George Casey, the top U.S. commander in Iraq, told reporters after Rumsfeld announced the troop cuts Friday that he hopes to recommend more reductions as early as next spring, assuming progress, such as formation of an Iraqi government.
But he added, "I don't have a goal for the end of 2006."
Rumsfeld announced that two Army brigades — one from the 1st Infantry Division in Kansas and the other from the 1st Armored Division now in Kuwait — will have their deployments to Iraq canceled. Casey said the total number of U.S. troops in Iraq will drop by about 7,000, to about 130,000 by March. This year's base level has been about 138,000.
"In this kind of war that we're fighting, more is not necessarily better," Casey said.
Caroline Wadhams, a senior national security analyst at the liberal Center for American Progress, said it's hard to tell whether this will ease public dissatisfaction with the war. But, she said, "even if the numbers don't decrease significantly, it's still better if they're not doing the high-visibility combat."
In remarks to U.S. troops in Fallujah and to Iraqi and American officials in Baghdad, Rumsfeld said the U.S. "footprint" must not be so large or intrusive as to "antagonize a proud and patriotic people, or to discourage the Iraqi people from taking initiative to run their own country."
Congressional Democrats praised the announcement and urged President Bush to go further. House Minority Leader Nancy Pelosi, D-Calif., said she hoped the reduction "will quickly be followed by others that will result in all U.S. combat forces being redeployed from Iraq next year."
Besides the force reduction, American officials have begun talking about shifting the roles of some U.S. combat troops to more behind-the-scenes tasks, such as advising Iraqi units.
"The coalition will continue to transfer responsibility for security operations to the Iraqi security force and place more emphasis on supporting Iraqi forces through training, support activities and counter-terrorist operations," Rumsfeld told several hundred soldiers, sailors and Marines in a Fallujah hall decorated for Christmas.
There were conflicting accounts by officials of how big the initial troop cut will be. Rumsfeld mentioned no specific number, saying only that the force would be cut by two brigades. His aides said privately that the number likely would be 5,000 or less, but Casey put it at 7,000.