A major row erupted over a report that suggested Tory leadership contender Andrea Leadsom was using her status as a mother to give her an advantage over Theresa May.
The energy minister said she wanted a retraction from The Times, while the newspaper published a transcript of parts of its interview with her.
May does not have children, and Leadsom's comments caused a huge backlash, including criticism from senior Conservatives Ruth Davidson and Sir Alan Duncan.
The controversy came after the Home Secretary urged her rival to sign a pledge committing to a clean campaign.
Leadsom has two sons and a daughter, and the newspaper's headline was: "Being a mother gives me edge on May -- Leadsom".
In the interview, Leadsom said May "possibly has nieces, nephews, lots of people. But I have children who are going to have children who will directly be a part of what happens next."
The energy minister said: "Genuinely I feel that being a mum means you have a very real stake in the future of our country, a tangible stake."
Asked to contrast herself with May, she said: "I see myself as one, an optimist, and two, a member of a huge family and that's important to me. My kids are a huge part of my life."
Leadsom said: "It means you don't want a downturn but, never mind, 10 years hence it will all be fine. My children will be starting their lives in that next 10 years so I have a real stake in the next year, the next two."
She also said: "I am sure Theresa will be really sad she doesn't have children so I don't want this to be 'Andrea has children, Theresa hasn't', because I think that would be really horrible."
As supporters of May lined up to attack Leadsom, the energy minister accused the newspaper of "gutter journalism" and "despicable and hateful reporting".
In a later statement Leadsom said: "I am beyond anger and disgust. The reporting of what I said is beneath contempt.
"In front of The Times correspondent and photographer, I made clear repeatedly that nothing I said should be used in any way to suggest that Theresa May not having children had any bearing whatever on the leadership election.
"I expect The Times to retract the article and the accompanying headline."
The interview came just days after the Home Secretary spoke about how her and husband Philip were affected by being unable to have children.
May told the Daily Telegraph she likes to keep her "personal life personal" but says that she and her husband "dealt with" the fact they couldn't have children and "moved on".
"I hope nobody would think that mattered," she said. "I can still empathise, understand people and care about fairness and opportunity."
MPs joined in to show support for May:
I am childless. I have nieces and nephews. I believe I - like everyone else - have a very real stake in our country. pic.twitter.com/urIPmRXfXE
-- Ruth Davidson (@RuthDavidsonMSP) July 8, 2016
I'm gay and in a civil partnership. No children, but ten nieces and nephews. Do I not have a stake in the future of the country? Vile.
-- Alan Duncan (@AlanDuncanMP) July 8, 2016
Andrea #Leadsom apparently thinks the c, 20% of women who won't have children are second-class citizens. Tories steering back to 1950s?
-- Natalie Bennett (@natalieben) July 9, 2016
Some of them believe an apology for May is due:
Either way, an apology is due.
-- David Gauke (@DavidGauke) July 8, 2016
-- Anna Soubry MP (@Anna_Soubry) July 9, 2016
..we need a PM w real experience at the highest level, proven competence, a safe pair of hands, a compassionate one nation Tory #TheresaMay
-- Anna Soubry MP (@Anna_Soubry) July 9, 2016
Leadsom's comments came as the Home Secretary urged her to sign up to an agreement to campaign within "acceptable" boundaries and not to work with other parties after the energy minister attracted the support of prominent Ukip figures.
The energy minister, who backed a vote to leave the European Union, has received the endorsement of both Nigel Farage and Brexit campaign Leave.EU, which is led by Ukip donor Arron Banks.
Leadsom told The Times: "I'm no Ukip sympathiser, they don't advise me, I don't know them, I've never even met Arron Banks.
"My big hope in this campaign is that when we leave the EU that Ukip will be a thing of the past. Hate crimes? It's absolutely appalling. I reject the premise that it's to do with the campaign."
May's "clean campaign pledge" commits the two candidates to:
1. Stick to the spending limits set by the Conservative headquarters
2. Not co-operate "in any way" with other political parties, their donors, members or active supporters
3. Do "everything in our power" to ensure that supporters' campaigning on social media is "in good taste"
4. Ensure the campaign stays within "the acceptable limits of political debate"
May used the interview with the Daily Telegraph to try to reassure Brexit-backing Tories that she was committed to leaving the EU, despite backing a Remain vote , saying: "If I am prime minister we will come out of the European Union and part of that will be control of free movement.
"But alongside that it's important to show how we can come through what will be I think some difficult times with a better, brighter future."
She added: "I am very clear that Brexit means Brexit. But I don't think we should see people as Brexiteers and Remainers now.
"We have a job to do in making the best deal we can in coming out of the EU and I am very clear that I will deliver Brexit."