Updated 1:50 p.m. ET, 11/18/2013
(CNN) – One day after their daughters' quarrel over same-sex marriage burst into the political spotlight, former Vice President Dick Cheney and his wife, Lynne, defended their daughter Liz Cheney, saying she loves and respects her sister Mary, who's married to a woman.
"This is an issue we have dealt with privately for many years, and we are pained to see it become public," the Cheneys said in a statement.
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"Since it has, one thing should be clear. Liz has always believed in the traditional definition of marriage," the Cheneys added, saying Liz Cheney has "always treated her sister and her sister's family with love and respect, exactly as she should have done."
"Compassion is called for, even when there is disagreement about such a fundamental matter and Liz's many kindnesses shouldn't be used to distort her position," the statement concludes.
The recent dispute started when Liz Cheney, who's making a GOP primary challenge for a U.S. Senate seat in Wyoming next year, said Sunday morning that she supports some rights for same-sex partners but does not favor the right for same-sex couples to legally wed.
"I love Mary very much. I love her family very much. This is just an issue in which we disagree," Cheney said on "Fox News Sunday."
Mary Cheney responded Sunday not long after the interview.
"Liz – this isn't just an issue on which we disagree – you're just wrong – and on the wrong side of history," she posted on Facebook.
Mary Cheney married her longtime partner, Heather Poe, last year in Washington D.C., which legalized same-sex marriage in 2009. The couple have two children together – a son, Samuel, and a daughter, Sarah. Liz Cheney did not attend the wedding.
Mary also shared a post by her wife, who said she was offended by Liz Cheney's latest comments:
I was watching my sister-in-law on Fox News Sunday (yes Liz, in fifteen states and the District of Columbia you are my sister-in-law) and was very disappointed to hear her say "I do believe in the traditional definition of marriage."
Liz has been a guest in our home, has spent time and shared holidays with our children, and when Mary and I got married in 2012 – she didn't hesitate to tell us how happy she was for us.
To have her now say she doesn't support our right to marry is offensive to say the least
I can't help but wonder how Liz would feel if as she moved from state to state, she discovered that her family was protected in one but not the other.
I always thought freedom meant freedom for EVERYONE.
Mary Cheney added: "Couldn't have said it better myself."
Responding to the comments, Cheney said Monday morning she treats her sister with compassion.
"I love my sister and her family and have always tried to be compassionate towards them. I believe that is the Christian way to behave," Liz Cheney told CNN anchor and Chief Washington Correspondent Jake Tapper.
Mary Cheney's sexual orientation was known during her father's first run in 2000 for vice president, though her family generally declined to discuss her personal life on the campaign trail.
Dick Cheney has said in the past that gays and lesbians should be allowed to marry, though like his daughter Liz he says regulations should be handled at the state level.
Tapper asked Cheney last month in an interview for CNN's "The Lead" whether the family disagreement might create awkward Christmas table conversation.
"My position on that issue is well known, I enunciated it in 2000 in a debate with Joe Lieberman. It hasn't changed. And I'll let my daughters speak for themselves," the former vice president said.
A majority of Americans voters–56%–say they support or would support allowing same-sex marriage in their state, while 36% oppose, according to a Quinnipiac University poll released in late September.
When considering Republicans' views only, the numbers flip. Fifty-eight percent oppose laws that legalize same-sex marriage, while 36% favor such laws.
The American Principles Fund, an independent political group defending Wyoming's incumbent GOP Sen. Mike Enzi, launched an ad last month attacking Liz Cheney for supporting some same-sex rights. Wyoming is one of 34 states with a constitutional amendment banning same-sex marriage.
The ad said she opposed a constitutional amendment to ban same-sex marriage and applauded the State Department's decision to grant benefits to same-sex partners of employees.
Cheney said Sunday that Enzi should "renounce" the ad for being "gutter politics." She defended her view on the State Department and explained that she opposed the constitutional amendment because she believes same-sex marriage decisions should be made on the state level.
"I don't believe we ought to discriminate against people because of their sexual orientation. If people are in a same-sex relationship and they want their partner to be able to have health benefits or be designated as a beneficiary on their life insurance, there's no reason we shouldn't do that," she said.
"I also don't support amending the Constitution on this issue. I do believe it's an issue that's got to be left up to the states. I do believe in the traditional definition of marriage."
When Cheney voiced her stance on same-sex marriage in late August, her sister took to Facebook to say Liz Cheney was "dead wrong" on the issue.
Asked about the dispute, White House spokesman Jay Carney said "This is a tough issue, and can be a tough issue, for a lot of families."
"It is a very American value to pursue equality for everyone," he added.
- CNN’s Jake Tapper and Kevin Liptak contributed to this report.