In a separate move, the four Democratic members of Wisconsin's congressional delegation asked U.S. Attorney General Eric Holder to recognize the marriages of more than 500 gay couples who recently wed in the state.
Both Burke and Walker were asked about their positions on gay marriage during separate events in Madison. Walker said Burke, who is running her first statewide campaign, needs a lesson in "Government 101" because she doesn't understand the governor "can't arbitrarily change the constitution."
But Burke, at a different event just a couple of miles away, said Walker had been vague regarding what she called a "critical issue."
"I think the people of Wisconsin would like to hear what the governor thinks," Burke said. "I know as governor I'm going to be clear where I stand on issues. It seems pretty political to me that he seems now to be waffling on whether he supports gay marriage or doesn't. And that is not being clear with voters about that."
Walker responded by repeating what he said on Thursday: He voted for the ban in 2006, but now his stance doesn't matter because the governor plays no role in changing the constitution. Only the Legislature, the state's voters or a judge can make changes, he said.
"The governor has no bearing on that issue," Walker said. "It's not a matter of changing positions."
A federal judge struck down Wisconsin's ban as unconstitutional on June 6, and said same-sex couples can get married in the state, but on Friday put that ruling on hold pending ongoing appeals from Republican Attorney General J.B. Van Hollen.
Clerks in all but a dozen of Wisconsin's 72 counties issued marriage licenses to same-sex couples during the week between U.S. District Judge Barbara Crabb's initial decision and her order stopping weddings.
Walker was outspoken in supporting the ban in 2005, when he was briefly a candidate for governor. He voted for the ban and had a history, both as a member of the state Assembly and as Milwaukee County executive, of opposing same-sex marriage and domestic partner benefits.
"It's a critical issue and the people of Wisconsin want to know where the governor stands," Burke said. "If I was governor I would want to see gay marriage go forward."
Recent polls in Wisconsin show public opinion is moving in favor of gay marriage. The most recent Marquette University law school poll released last month showed 55 percent of registered Wisconsin voters favor allowing gays and lesbians to marry legally. That same poll showed Walker tied with Burke among registered voters.
Also on Monday, U.S. Sen. Tammy Baldwin and U.S. Reps. Ron Kind, Gwen Moore and Mark Pocan sent Holder a letter asking him to recognize the marriages that had already been performed. They noted Holder previously recognized marriages performed in similar circumstances in Utah and Michigan.
Holder's spokeswoman, Dena Iverson, said the Department of Justice is monitoring developments in Wisconsin. She declined further comment.
Federal recognition would allow couples to file their federal tax returns jointly, qualify for Social Security survivor benefits and help foreign spouses with immigration. The spouses of federal workers and service members could qualify for health, retirement and other benefits.
Wisconsin has a five-day waiting period for marriage licenses. Some same-sex couples who applied for licenses but didn't receive them before Crabb halted ceremonies on Friday sought refunds Monday.
Johnson reported from Milwaukee. Associated Press writer Todd Richmond contributed to this story from Madison.