This is a yes when the question starts with "can" and a no when it starts with "will." It's possible to keep tabs on this and put a stop to it, but it'll require dedication from cycling that is unlikely to happen. Lance Armstrong can pretty much put any "give your winnings back" lawsuit to rest by asking for proof that the sport wasn't overrun with people doing exactly what he was.
With the recent outing of Lance Armstrong as cycling's poster child, the doping culture will certainly be challenged. If someone as highly regarded as Lance Armstrong can fall, surely those with lesser status will be forced to take caution. Many of the past Tour de France winners have had their winnings invalidated, and the pressure to avoid doping will increase as new contestants are witness to the increased scrutiny placed on them.
Being the best cyclist in the world brings a lot of fame and fortune along with it. For as long as there has been athletic events, there have been athletes looking for ways to circumvent the rules to gain an edge over the competition. I don't see this problem resolving itself.
Cycling is an international sport, and it is very difficult to regulate things at the global level. The fact is that cycling rewards the fastest people. It rewards them so much so that it has become worth the risk of cheating and doping. Whenever we find a test for a certain drug, there will always be another, newer drug out there.