The UCI, cycling's world governing body, on Monday ratified the sanctions recommended by the United States Anti-Doping Agency, who concluded Armstrong and his United States Postal Service team ran "the most sophisticated, professionalised and successful doping programme that sport has ever seen".
The 41-year-old was stripped of his seven Tour titles, won between 1999 and 2005, and banned for life.
Speaking at the launch of the 2013 Tour, the 100th edition of cycling's most prestigious race which takes place next summer, Wiggins said: "I think there is a lot of anger from most people within the sport, it is a sport I love and have always loved.
"It is a shame that cycling is being dragged through this again really, not a shame that he has been caught - when you get older you start to realise Father Christmas doesn't exist and it is the same with Lance.
"But it is a shame that us riders here now, we are the one picking the pieces up and having to convince people."
Team Sky's Wiggins, who became the Tour's first British winner in July, does not feel the latest doping revelation to hit cycling will detract from his own success.
"It doesn't take away from my victory but as winner of the Tour de France you are subjected to a lot of other things, it goes beyond sport," he said.
"It is out there now, hopefully this sport can move forward - the route has been announced today [Wednesday], cycling isn't like that any more and we (the riders) are evidence of that."
Armstrong retired after his 2005 Tour win, before returning to compete in the 2009 and 2010 races.
Tour director Christian Prudhomme is adamant the race and the sport can survive the scandal which has discredited both.
"The Tour belongs to those who love it," Prudhomme said.
"It will be stronger, stronger than doping and stronger than cheating, the enemy is the doping, it isn't the cycling.
"There is no war between cycling and other disciplines, just the good on one side and the bad, the cheats on the other."
Jean-Etienne Amaury, the president of Tour organisers Amaury Sports Organisation, added: "For several years ASO has been campaigning to fight the plagues that goes against the spot - it has to be reinforced.
"The problem is the presence of dishonest people, some have been caught by justice, others still have to be."
Mark Cavendish, the 2011 world champion, earlier urged Armstrong to come clean.
The Omega Pharma-QuickStep sprinter told BBC Sport: "It's not fair on me having to answer these questions. If you've done something, confess.
"That anyone can damage the sport I love right now, it's frustrating."
Wiggins agreed with his compatriot but does not feel the admission will be forthcoming.
When asked whether he thought Armstrong admitting to the charge would help the situation, Wiggins said: "I think so, definitely.
"But he is a stubborn man and I don't think he is ever going to confess, he has too much to lose."
Team Sky principal Dave Brailsford preferred to focus on the future.
Brailsford, who is also British Cycling performance director, told Sky Sports News: "(Doping) is certainly dominating the headlines at the minute but I think the sport has got to move forward and this is about the future.
"We can't change what has happened in the past but what we can try and be is agents for change in the future.
"This sport needs to regain the trust that it has lost because of the past but I think when you look at next year's Tour it gives everyone an opportunity to focus on what we can change for the future.
"We can't change the past but we can change what we do today, tomorrow and by the time we get to next year's Tour."
The UCI management committee will meet on Friday to discuss whether Armstrong's titles and prize money will be re-distributed, following which the International Olympic Committee will make a decision on the bronze medal Armstrong won in Sydney in 2000.
Armstrong has repeatedly denied wrongdoing and refused to co-operate with the USADA investigation.
Both he and WADA could yet take the case to the Court of Arbitration for Sport.
Armstrong did receive support, though, with former Astana team-mate Alberto Contador standing up for him.
The Spaniard, who rode with Armstrong in 2009, was quoted by Cycling News as saying: "In my opinion, at certain times, there has been a total lack of respect for Lance.
"From my point of view, he has been humiliated and lynched. He has been destroyed. If cycling is popular in the United States, it's thanks to him. If they know the Tour, it's thanks to him."
Contador, who himself has served a doping suspension, added: "People talk about Lance but there is no new proof against him, nothing.
"They've relied on testimony that dates from 2005. I respect the choice of every rider, but I would have preferred if they had been a bit earlier."