After Johnson's announcement not to contest, May is now the clear favourite as a unifying candidate after a referendum campaign that exposed deep divisions within the party.
The former London Mayor and top Brexit campaigner was seen as her main rival to replace Cameron who had announced to step down after the referendum results in favour of Britain's exit from the European Union (EU) last week.
She underlined that "Brexit means Brexit" as she announced that she will be running for party leadership contest, which is set to conclude by September 9.
"Our country needs strong proven leadership to negotiate the best possible terms for the UK leaving the EU. Brexit means Brexit. The campaign was fought, the vote was held, turnout was high and the public has given its verdict," the 59-year-old Conservative party leader said, adding, "We need leadership that can unite our party and our country".
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She spoke about invoking Article 50, the formal procedure for leaving the EU, which Cameron has left for his successor to do.
"Article 50 should not be invoked before the end of the year," she said.
She argued that under her leadership the Conservative Party will come back together, not just for Remain or Brexit, but for the whole country.
Calling for an "open contest", she added: "Whether it's a woman or a man it's about the qualities of the people doing the job."
If May succeeds in her bid to be the Prime Minister, she will become Britain's second female Prime Minister after Margaret Thatcher, someone she has been inevitably compared with due to her no-nonsense political demeanour and smart wardrobe.