"I love my daughters. They love their daughter. I was trying to say something positive about the way strong families deal with this issue," Kerry said in a statement issued in Las Vegas, where he was speaking at an AARP convention.
Cheney's comments followed those of his wife, Lynne, who at a rally outside Pittsburgh on Wednesday night accused Kerry of "a cheap and tawdry trick."
In Florida today, the vice president called Kerry "a man who will say and do anything in order to get elected."
"And I am not speaking just as a father here — though I am a pretty angry father — but as a citizen," Cheney said, going on to discuss Kerry's position on the war in Iraq, according to a transcript of the remarks. He made no other reference to Kerry's statement.
Several sharp responses followed today over Kerry's remarks about Mary Cheney, an official in her father's campaign. Kerry spoke in answer to a question from debate moderator Bob Schieffer about whether homosexuality is a choice.
Kerry said: "We're all God's children, Bob, and I think if you were to talk to Dick Cheney's daughter, who is a lesbian, she would tell you that she's being who she was. She's being who she was born as. I think if you talk to anybody, it's not a choice."
Missouri Rep. Dick Gephardt, who also has a lesbian daughter, said today that he was "perplexed" by the Cheneys' reaction.
"I think what John Kerry said was very kind and nice and appropriate. He was complimenting the Cheney family on how they've dealt with this situation," Gephardt told reporters in a Kerry campaign conference call set up to promote a new television ad on job losses in Ohio.
Lynne Cheney did not see it that way.
"Now, you know, I did have a chance to assess John Kerry once more and now the only thing I could conclude: This is not a good man," she said at a rally outside Pittsburgh. "Of course, I am speaking as a mom, and a pretty indignant mom. This is not a good man. What a cheap and tawdry political trick."
Like her husband, she did not talk specifically about Kerry's statement about her daughter.
The wife of vice presidential candidate John Edwards said Mrs. Cheney "overreacted."
"She's overreacted to this and treated it as if it's shameful to have this discussion. I think that's a very sad state of affairs. ... I think that it indicates a certain degree of shame with respect to her daughter's sexual preferences," Elizabeth Edwards said in an interview today with ABC Radio.
The vice president expressed no objection when Edwards brought up Mary Cheney during their debate. Edwards expressed "respect for the fact that they're willing to talk about the fact that they have a gay daughter, the fact that they embrace her. It's a wonderful thing."
Cheney thanked his opponent for the "kind words he said about my family and our daughter. I appreciate that very much."
In the debate Wednesday, Bush said he did not know whether homosexuality was a choice. He and Kerry spoke of their belief that marriage is the union of a man and a woman, but the president supports a constitutional amendment banning gay marriage and Kerry does not.
The executive director of the Log Cabin Republicans, Patrick Guerriero, said in a statement that Kerry could have made his point without citing Mary Cheney.
"However, this shouldn't distract us from the fact that President Bush, Karl Rove and other Republicans have been using gay and lesbian families as a political wedge issue in this campaign," Guerriero said.
Associated Press contributed to this report.