The six-time Olympic champion told his 450,000 Twitter followers it was hugely frustrating having to defend his sport against the "greed and deception" of a small minority.
The 36-year-old said he had been inundated with questions following Armstrong's face-to-face with Oprah Winfrey, which was broadcast to a global audience on Thursday.
During the interview, the former Tour De France winner admitted to taking banned substances while cycling competitively.
Despite Armstrong's apology hitting headlines around the world this week, Sir Chris said it was time to look to the future and move on from the scandal.
Sir Chris, from Edinburgh, said: "I've been inundated with questions online and at the London Bike Show today about Armstrong's interview.
"My opinions of him and the damage he has done to cycling haven't changed in the last 24 hours, however I believe we need to look to the future and move on.
"It's hugely frustrating to have to defend your sport because of the greed and deception of a small minority.
"My team-mates and I will keep doing what we have always done; compete clean and try to win gold medals to show the next generation that it is possible."
Sir Chris, who won two gold medals at London 2012, has been an outspoken and fierce critic of drug cheats in the past.
After Armstrong was stripped of his seven Tour de France titles, Sir Chris said there was "nowhere to hide for the cheats."
While Armstrong's confession and apology will go some way to aid his recovery from the scandal, suspicion, anger and a cloud of unanswered questions hung in the air following his appearance.
Nicole Cooke, the 2008 Olympic road race champion, branded the prime-time interview a pantomime.
Armstrong admitted drug-taking but has not, so far, named names or revealed the depth of the corruption that allowed him to avoid detection for so long, Cooke noted.
"Lance Armstrong should have been taken to a court, not to an Oprah Winfrey sofa," she said.
Some cycling and anti-doping officials suggested Armstrong, now a self-confessed liar and bully, was still lying.
After years of fierce denials, the 41-year-old Texan admitted using the blood-boosting agent EPO, as well as taking testosterone, human growth hormone and the steroid cortisone. He also admitted blood doping.
He has been banned from sport for life and stripped of all his Tour titles – plus the bronze medal he won at the 2000 Sydney Olympics. The cyclist has been stripped of all his results from August 1, 1998.
However, he denied to Winfrey that he had doped during his comeback from retirement in 2009 and 2010.
British Cycling president Brian Cookson said he believed Armstrong seemed sorry for getting caught rather than for using drugs.
He said: "It always has been cheating. It's always been against the rules.
"It's always been banned substances – there's a clue in the name, as they say.
"He knew very well what he was doing. I don't accept the argument that he was going up on a level playing field.
"I think he's got a lot of explaining to do, he's got a lot of recompensing to do. He's got a long way to go yet before he earns any degree of forgiveness as far as I'm concerned."
On Armstrong's claim he stopped doping in 2005, the World Anti Doping Agency president John Fahey said: "The evidence from USADA [the US Anti Doping Agency] is that Armstrong's blood tests show variations in his blood that show with absolute certainty he was doping after 2005. Believe USADA or believe Armstrong? I know who to believe."
USADA chief executive Travis Tygart said: "His admission that he doped throughout his career is a small step in the right direction.
"But if he is sincere in his desire to correct his past mistakes, he will testify under oath about the full extent of his doping activities."
Armstrong said he believed doping was part of the process required to win the Tour and he did not feel he was cheating at the time. He considered that he was competing on a level playing field.
Armstrong said he did not fear getting caught, but that now all the fault and blame for the situation should lie with him.
He said his cancer fight in the mid 1990s gave him a win-at-all costs attitude.
He said he was a bully who turned on people he did not like but he would now co-operate with official inquiries into doping in cycling. While confessing to Winfrey, he admitted he might appear hard to trust, noting: "I'm not the most believable guy in the world right now."