The energy minister said she wanted a retraction from The Times, while the newspaper published a transcript of parts of its interview with her.
Mrs May does not have children, and Mrs 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.
Mrs 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, Mrs Leadsom said Mrs 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 Mrs 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.”
Mrs 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 Mrs May lined up to attack Mrs Leadsom, the energy minister accused the newspaper of “gutter journalism” and “despicable and hateful reporting”.
In a later statement Mrs 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.
TRANSCRIPT FROM THE ANDREA LEADSOM INTERVIEW WITH THE TIMES
The Times has published an extract from its interview with Andrea Leadsom that sparked a major row over how her comments on motherhood and her rival Theresa May were reported.
Interviewer: During the debates you repeatedly said as a mum. Do you feel like a mum in politics?
Mrs Leadsom: “Yes. 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 but genuinely I feel that being a mum means you have a very real stake in the future of our country, a tangible stake.
“She 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.”
Interviewer: “So it really keeps you focussed on what are you really saying?”
Mrs Leadsom: “It means you don’t want a downturn but never mind ten years hence it will all be fine, my children will be starting their lives in that next ten years so I have a real stake in the next year, the next two.”