World champion Cadel Evans of Australia took the yellow jersey at the end of the stage, with overnight leader Sylvain Chavanel - like Armstrong - falling behind the top contenders on the tough La Ramaz climb.
The Texan crossed the finish line nearly 12 minutes behind Schleck during the 117.4-mile run from Station des Rousses to the Morzine-Avoriaz ski resort.
"My Tour is finished," said Armstrong, who holds a record seven Tour victories and is riding in what he says will be his last try in cycling's showcase event - at age 38. He plunged to 39th place overall.
Armstrong said he had a "very, very bad day," but will continue the three-week race that ends July 25 in Paris.
"No tears from me," he said. "I've had a lot of years here where it's been very different, so I'm not going to dwell on today."
Schleck showed he's one of the world's best climbers by surging to the head of the pack with less than a kilometer left and winning a two-man sprint ahead of Samuel Sanchez of Spain.
Schleck, the Saxo Bank team leader, clocked 4 hours, 54 minutes, 11 seconds, while Sanchez had the same time.
Dutch rider Robert Gesink was third, 10 seconds back - as were defending champion Alberto Contador of Spain, and Evans.
Armstrong was 11:45 back, in 61st place.
Overall, Evans leads Schleck by 20 seconds and Contador by 61 seconds. The Australian is a two-time Tour runner-up, and Schleck was runner-up last year.
"I didn't want to lose too much time to Andy, but I'm happy to finish with Contador today," Evans said. "And obviously, sitting here with the yellow jersey isn't a bad position to be in."
Armstrong is 13:26 back, having begun the day in 14th place, 3:16 behind Chavanel.
With about 19 kilometers left, shortly after La Ramaz, Armstrong was held up by a rider who crashed in front of him. Though the Texan rider did not fall, he slowly got back on his bike, in seeming exasperation at his third mishap.
Earlier, with less than 51 kilometers to go, Armstrong tumbled to the asphalt while trying to negotiate a roundabout along with the pack. His team quickly got him another bike and he returned to the race with the back of his jersey and his bib numbers torn.
"I clipped the pedal and then my tire rolled off," Armstrong said. "Then, the next thing I know, I was rolling along the ground, at 60-65 kilometers an hour."
He said it was just too tough to clamber back after that.
Armstrong then struggled up the steepest portion of La Ramaz, losing crucial seconds to the main overall contenders, including Schleck, Contador and Evans.
"Obviously the Tour's finished for me," Armstrong said. "But I can try and win stages, try and help the team, really try and appreciate my time here, and the fact that I'm not coming back."
He later tweeted. "When it rains it pours I guess. Today was not my day needless to say. Quite banged but gonna hang in here and enjoy my last two weeks."
An earlier incident involving Armstrong and occurred at about the six-kilometer mark. Armstrong narrowly averted a spill but still lost time, while Evans fell, ripping his shorts and scraping his elbow and knees.
On Monday, riders will get the first rest day of the race, which finishes on July 25 in Paris.