The demand by Roy Blunt of Missouri, a member of the Senate Republican leadership, is the first by a lawmaker and comes just about 24 hours after McDonald made his remark to a group of reporters Monday.
"When you go to Disney, do they measure the number of hours you wait in line? Or what's important?" McDonald said at a Christian Science Monitor breakfast in Washington
. "What's important is what's your satisfaction with the experience."
Blunt, who is running for re-election in what could be a tight race, issued a statement slamming McDonald as being "right out of Never Never Land."
"I call on him to resign because it's clear he cannot prioritize getting our veterans the health care they deserve and have earned in a timely manner. Dismissing wait times when veterans can often wait months for an appointment is negligent and a clear sign that new leadership is needed at the VA," Blunt said.
McDonald stopped short of apologizing for the remarks Tuesday, telling MSNBC's Andrea Mitchell
he was glad to have the "opportunity to correct" the record.
"If I was misunderstood, if I said the wrong thing, I'm glad that I have the opportunity to correct it," he said. "I'm only focused on one thing, and that's better caring for veterans. That's my job, that's why I'm here."
Also Tuesday, McDonald's comparison drew the ire of several House Republican leaders who spoke to reporters at a weekly news conference.
Asked if McDonald should resign, House Speaker Paul Ryan, R-Wisconsin, didn't go that far, but he called McDonald's comments "disgusting" and said he needed to clarify them. Ryan said the remarks reflect a "culture of indifference at the VA," adding that no one has been held accountable at the agency for the scandal.
One key Democratic senator told CNN he was "troubled" by McDonald's comments but praised the secretary as "responsive" and focused on good care for veterans.
"I am troubled by the words but I hope there is better intent behind them and they do not reflect a broader attitude that delay and lack of accountability are okay," said Sen. Richard Blumenthal of Connecticut, the top Democrat on the Committee on Veterans' Affairs. "He's been very responsive. He knows that care has to be high quality and timely and that's been a goal. I look forward to talking to him about what he meant."
Blumenthal said he is pushing bipartisan legislation, known at the Veterans First Act, that will help address cultural problems at the VA. He hopes it will be considered before the summer recess.
Co-sponsored by the committee's chairman, Republican Sen. Johnny Isakson of Georgia, the bill will "give the VA the tools to fire bad actors, will prohibit bonuses for employees accused of wrongdoing, and will institute protections for whistleblowers," according to a committee statement.
McDonald took over the troubled department about two years ago after former Secretary Eric Shinseki was forced out of the job because of persistent problems of long waiting periods for veterans seeking medical services from the department.