Australian Jaimie Fuller, whose Skins company is a partner of the elite Rabobank team and Cycling Australia among others, has written an open letter to McQuaid warning of catastrophic consequences of the body's "inertia".
The UCI has yet to rule on the US Anti-Doping Agency's report into Armstrong, which alleges that the now retired seven-times Tour de France champion achieved his success on the back of a highly sophisticated doping scheme.
"I'm just devastated and horrified by what has happened," Fuller said. "The initial response of the UCI to USADA was defensive and questioning not cooperative and the longer the subsequent silence goes on it's getting much worse for the sport of cycling.
"I'm an optimist and I would desperately hope that common sense would see the light of day.
"(But) if Mr McQuaid and his cohorts are not prepared to say what they are going to do about this systemic issue, to rip the scabs off of what has happened, there is no question they should resign."
The UCI, who can either confirm Armstrong's life ban and strip him of his seven Tour titles or take the matter to the Court of Arbitration for Sport, have until October 31 to rule on the case.
Sportswear manufacturer Nike Inc and beer maker Anheuser-Busch have both decided to terminate their relationships with Armstrong in the wake of the report.
Fuller warned the commercial impact would go beyond the American and could dwarf the damage the sport suffered in the wake of the Festina team doping scandal of 1998.
"It happened at the end of the 90s with the Festina affair, to the point where German television refused to show the Tour de France," Fuller said.
"There's no question that there was a commercial hit that happened to the sport of cycling. I genuinely believe that what we're seeing now is way worse.
"When Armstrong was winning, when he was doing unbelievable things, the glow he cast across the whole sport was fabulous. Now, we're going to see the exact reverse."
Fuller said the company, which produces therapeutic compression clothing for athletes, would have to reconsider its association with cycling if the UCI failed to act.
"I'm really most concerned with the impact on the popularity of cycling," he said.
"That has both an emotional impact on those who love the sport as well as a commercial impact on we businesses who are involved in the sport.
"Our whole brand is built on the philosophy of integrity in sport. We enable relationship with lead athletes around the world on those grounds and we also enter into relationships with consumers with those beliefs."