Her campaign chief, Tim Loughton MP, made note of the fact that the next prime minister will be the country's first female leader since Margaret Thatcher, who led the United Kingdom from 1979 to 1990.
The race to become leader of the Conservative Party comes after David Cameron stepped down after losing the June 23 European Union referendum.
Results are expected September 9.
The contest between Mrs May and Mrs Leadsom will mean that the country will have a female premier for just the second time, following Margaret Thatcher.
Mrs Leadsom was a prominent Brexit-backer and hopes that her support for leaving the European Union will win her votes from Eurosceptics in the Tory grassroots.
Ms Davidson told the BBC's Newsnight programme that Mrs May had the "vast majority" of the parliamentary party behind her, and had "huge support" in all parts of the United Kingdom, including among the Scottish Conservatives. The victor will replace Prime Minister David Cameron who announced his resignation after Britons voted on June 23 to leave the European Union.
The final vote is in the hands of 150,000 Conservative Party members. The justice secretary torpedoed the campaign of former mayor of London Boris Johnson at the last minute last Thursday, withdrawing his support and declaring that he would stand in his place.
She paid tribute to Mrs Leadsom's "guts" in putting her name forward for leader, and said she hoped the leadership contest would see a "big, broad debate" being held across the country.
Britain's next prime minister nearly certainly will be a woman. The victor of the second round will take charge as the PM and party leader and will be announced on September 9. Johnson has since declared his support for Leadsom. "Isn't this the new sort of Tory party?"
"Andrea Leadsom is undoubtedly the outsider, but she will play up her Brexit credentials", said Al Jazeera's Barnaby Phillips on Thursday in Westminster.
Mohamed El-Erian, chief economic adviser to Europe's largest insurer Allianz, told Reuters news agency on Thursday that the pound could sink towards parity with the dollar unless politicians got a grip.
Leadsom, 53, who backed the "leave" campaign in the referendum, entered Parliament in 2010 after a career in financial services.