"Job well done," the president declared.
Obama addressed roughly 2,000 troops after meeting privately with the full assault team who executed the raid on bin Laden's compound early Monday. Their identities are kept secret.
Speaking to a sweltering hangar full of cheering soldiers, Obama said: "The terrorist leader that struck our nation on 9/11 will never threaten America again."
Fresh warnings emerged, though, underscoring Obama's caution that the fight against terrorists still rages.
The Afghan Taliban said the death of bin Laden would only boost morale of insurgents battling the U.S. and its NATO allies.
Soldiers at Fort Campbell were careful not to celebrate bin Laden's death, voicing instead a sense of professional pride for the work of the commandos.
"We're not done," said Maj. Luis Ortiz, who was at Bagram Air Base in Afghanistan when Obama visited the troops there last December. "We cut off the head of the snake, but the snake is still wiggling around."
Obama called the bin Laden raid one of the most successful intelligence and military operations in America's history, and said he had to come to extend personal thanks.
Vice President Joe Biden joined Obama in a briefing with the mission members and then emerged to put it bluntly: "We just got to spend time with the assaulters who got bin Laden."
Obama's appearance culminated a weeklong response to the demise of the long-hunted al-Qaida leader, from the White House to ground zero in New York to Fort Campbell, home of the famous 101st Airborne Division. The division has been integral to Obama's war plan in Afghanistan, and many of its combat teams have returned recently from tours of duty.
The week gave a political and emotional lift to the president; in turn, he called for the unity that has eluded him in divisive Washington for most of his term.
"This week has been a reminder of what we're about as a people," Obama said. "The essence of America, the values that have defined us for more than 200 years, they don't just endure -- they're stronger than ever."
With his comments, Obama offered a counterpoint to a growing cry within his party and even among some Republicans that the time has come to withdraw from Afghanistan. Obama will start bringing troops home this summer as promised but has signaled no change in mission.
At Fort Campbell, the president and vice president first met with the men who raided the compound itself, probably including those who killed bin Laden.
Obama was then briefed on how the operation was carried out, by those who coordinated the attack from command centers in Afghanistan and in other undisclosed parts of the region.
That team was headed by Vice Adm. William McRaven, a Navy SEAL himself and head of the military's elite counterterrorism unit, the Joint Special Operations Command. Obama and Biden then met with the entire SEAL team unit that carried out the raid -- both the two dozen troops who stormed the compound, and roughly the same number who circled above as backup, in case the SEALs on the ground met overwhelming force.
The president also met with air crews from the 160th Special Operations Aviation Regiment, known as the Night Stalkers, who flew the SEALs to the mission, and Green Berets from the 5th Special Forces Group.
The president awarded the units a Presidential Unit Citation -- the highest such honor that can be given to a unit -- in recognition of their extraordinary service and achievement.