In his first major speech of the Tories' long leadership campaign, Mr Davis said the party had to emulate Bob Geldof's success at the Live8 concert by showing the same idealism.
"The contrast with Westminster's politics is striking. I don't agree with every policy proposed by Make Poverty History, although I share its sentiment. But I do think that we need a new Tory idealism and an uplifting vision of a better Britain," he said.
Addressing the Centre for Policy Studies, Mr Davis emphasised his upbringing on a council estate and made a coded criticism of the Eton-educated Mr Cameron, the 38-year-old shadow Education Secretary, who is emerging as his main rival when Michael Howard steps down before the end of the year.
Highlighting his personal credentials to lead a "one nation" Conservative Party, Mr Davis said: "We shouldn't be in politics to defend privilege. At its greatest, the Conservative Party has spoken for one nation, for the many not the few.
"I want us to be the champion for the victims of state failure, those without hope and opportunity under the current system. We shouldn't be in politics to accept the status quo."
Mr Davis, the shadow Home Secretary, rejected Mr Cameron's and Mr Howard's view that tax cuts were not a "silver bullet" that would restore the party to power.
He warned that "accepting the high-tax, high-spend terms of the debate set by Gordon Brown is certainly a bullet to the heart of electoral success".
He insisted that previous Tory governments had boosted tax revenues by cutting tax rates.
"Achieving a low-tax economy isn't about slashing state spending," he said. The poor, not the rich, ended up paying taxes. Low taxes were good for growth and helped to support stronger families and create a stronger society.
Mr Davis said: "We cannot credibly promise to change Britain just by being better ministers or managers than Labour.
"Timid politics, a few tweaks here and there to Labour's approach, will not address the cause and depth of Britain's problems. Neither will it persuade people that we have the ambition for our country to deserve office."
Mr Cameron's allies shrugged off the veiled criticism, saying that what mattered was not a candidate's background but their policies. They got a boost when Greg Barker quit the Tory whip's office to join the Cameron campaign.
He said: "I believe that David Cameron has got a lot to offer the country and has the ability to connect with a new generation. He has conviction and determination and would bring aspiration back into British politics."
Michael Ancram, the Tories' deputy leader, who may enter the leadership race, will issue a blunt critique of the party's problems today. He will say in a speech: "The truth is that since Margaret Thatcher the Conservatives as a party have lacked character, coherence and context. Despite often well-worked policies, there has been no real vision, no real sense of mission, and only a limited understanding of how our country has changed.
"It is agreeing principles that must now be the priority. If we fail to take this opportunity, then our party will remain in the political wilderness."
He will oppose plans for Tory members to lose their right to choose the party leader, saying that the franchise should be extended not reduced.
Born to lead?
By Genevieve Roberts
* Michael Howard, 2003-, father a shopkeeper, attended a state school and Cambridge.
* Iain Duncan Smith, 2001-3, son of an RAFofficer, naval training school and Sandhurst.
* William Hague, 19972001, father a businessman, comprehensive school and Oxford.
* John Major, 1990-97, son of a showman, grammar school, left at 16.
* Margaret Thatcher, 19751990, daughterof a grocer, grammar school and Oxford.
* Edward Heath, 1965-1975, son of a carpenter, state grammar, Oxford.
* Sir Alec Douglas-Home, 196365, son of a Scottish earl, Eton and Oxford.
* Harold Macmillan, 19571963, son of an earl, Eton and Oxford.
* Sir Anthony Eden, 1955 1957, son of landowners, Eton and Oxford.
* Sir Winston Churchill, 19401955, son of a politician, boarding schools and Sandhurst.