The US Anti-Doping Agency has removed the seven Tour de France titles that Armstrong won between 1999 and 2005, although the International Cycling Union (UCI) can still appeal against the decision.
"What we want is that there is no winner," said Prudhomme.
"We cannot be indifferent to what USADA has unmasked this week, it's a damning picture that's been drawn.
"As astonishing as it may seem, the Tour de France is not the master of its record books. That goes through the UCI rather than the race organisers."
Armstrong, who has always denied doping, has not contested the charges brought against him by USADA.