McQuaid had branded Hamilton and Floyd Landis as "scumbags" and accused them of trying to be heroes. Both testified against Lance Armstrong, having previously been disgraced by failed drug tests.
Hamilton's response came on Tuesday after World Anti-Doping Agency chief John Fahey said officials in charge of the sport at the time of the Armstrong affair had to take some of the responsibility.
McQuaid did not take over as UCI president until after Armstrong won the last of seven Tour de France titles, but his comments about Hamilton have provoked an angry response.
The UCI 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".
Armstrong, who could be forced to repay his Tour de France prize money and bonuses, removed reference to his Tour de France titles from a profile on his Twitter account.
Armstrong was banned for life and all his results from August 1, 1998 removed, including his wins at the Tour from 1999 to 2005.
The UCI management committee will meet on Friday to discuss the "exact sporting consequences" of the decision, including whether the titles and prize money will be re-distributed.
But UCI president McQuaid's comments about Hamilton have led to the rider issuing a statement in which he said: "Pat McQuaid's comments expose the hypocrisy of his leadership.
"Instead of seizing an opportunity to instil hope for the next generation of cyclists, he continues to point fingers, shift blame and attack those who speak out, tactics that are no longer effective. Pat McQuaid has no place in cycling."
McQuaid said on Monday: "Another thing that annoys me is that Landis and Hamilton are being made out to be heroes.
"They are as far from heroes as night and day. They are not heroes, they are scumbags. All they have done is damage the sport."
McQuaid, UCI president since 2006, rejected calls for his resignation over perceived failures by the world governing body over the Armstrong affair.
McQuaid insisted the UCI has "nothing to hide" over a donation of more than US dollars 100,000 (£62,000) made by Armstrong in 2002, denying it was connected to any cover-up of a positive test.
"Don't try to make the connection between the suspicious test and the donation. There were no positive tests from him," said McQuaid.
"It's certainly not a resignation issue."
WADA president Fahey on Tuesday claimed the sport would only regain credibility when the senior officials on watch take responsibility for the scandal.
Fahey told Fox Sports Australia: "They clearly have to take the blinkers off, look at the past, examine the people who are there, ask themselves the questions: 'Are those same people still in the sport and can they proceed forward with those people remaining?'.
"I don't think there's any credibility if they don't do that and I think they need to get confidence back into the sport so that its millions of supporters around the world will watch and support the sport going forward."
Tour de France director Christian Prudhomme believes the race must not have a victor in the Armstrong years.
Prudhomme said: "The formal decision has to be taken by the UCI but for us, we must have a clean record. This period must be marked by the absence of winners.
"The UCI rules are clear. When a rider is disqualified, he must pay the prize money back."
Five-time Tour de France winner Miguel Indurain believes Armstrong is innocent and that he will fight to clear his name.
"Even now I believe in his innocence. He has always respected all the regulations. He has won all the cases he's had," Spaniard Indurain told Radio Marca.
"I think he will come back and appeal and try to show that he played fair for all those years."
British Cycling president Brian Cookson, who is a member of the UCI management committee, said: "While this is painful for those of us who love cycling, I believe this is a necessary process and one from which the sport will emerge stronger.
"Cycling must act on the lessons from USADA's thorough investigation, which has brought light into a dark era in our sport."
Meanwhile, Armstrong has been asked to repay a disputed bonus payment.
The 41-year-old received a bonus of five million US dollars after a legal battle with SCA Promotions, who had declined to pay the sum in the belief the American had doped to win his seven Tour titles.
Jeff Tillotson, lawyer for SCA Promotions, told Press Association Sport: "It is inappropriate and improper for him to keep those bonus amounts and we will be demanding them back and pursuing appropriate legal action if he does not return them."
The International Olympic Committee will await Friday's UCI meeting and further information before a decision is made on the bronze medal Armstrong won in Sydney in 2000.
Armstrong, who battled back from cancer to return to professional cycling, has repeatedly denied wrongdoing and refused to co-operate with the USADA investigation.
Armstrong on Tuesday removed '7-time Tour de France champion' from his Twitter profile in his first response to being stripped of his seven titles for doping offences.
His profile said on Tuesday: "Raising my five kids. Fighting Cancer. Swim, bike, run and golf whenever I can."
Both Armstrong and WADA could yet take the case to the Court of Arbitration for Sport.