Mr Johnson said that Mrs May's Chequers blueprint - which ties Britain to a common rulebook with the EU for trade in goods - would be "politically humiliating for a £2 trillion economy" and would subject the United Kingdom to European directives and rules.
May's public battle with Johnson comes only two weeks before she must return to Brussels in a bid to break the Brexit deadlock and reach a deal.
"That is, in fact, a point on which he has to apologise, I think because we can not take it as such a thing he said". "That is not what we voted for", he said.
To cheers and applause, Mr Johnson declared: "This is not pragmatic, it is not a compromise".
The reaction to Boris arriving was a far cry from Prime Minister Theresa May's entrance when she turned up for Home Secretary Sajid Javid's immigration speech, which was significantly more subdued.
Asked what impact it would have on Mrs May's keynote speech tomorrow, Mr Davis said: 'It's a different speech, she's the prime minister, she's got to give a serious speech about the future of not just Brexit but all the other elements of the domestic strategy - which of course Boris was talking about too.
In a speech entitled "Our Future Is In Our Hands", she will tell delegates that countries across the world stand ready to trade with the UK.
As Mrs May celebrated her 62nd birthday, Mr Johnson was pictured jogging through a field near his Oxfordshire home, in a photo apparently created to mock the PM's famous memories of "running through wheatfields" as a mischievous schoolgirl.
Mrs May added that it was "frustrating" to hear Mr Johnson dismiss the so-called "backstop" plan to avoid a hard border on the island of Ireland.
Her broadside came ahead of her conference address on Wednesday at which Mrs May, who is facing heavy criticism from within the party for her Chequers plan for Brexit, will attempt to stamp her authority over her fractious and divided party. "We need to support the Prime Minister, but we're out to change the policy". She has insisted that her strategy is the only option for the country and that rejecting it would cause Britain to leave the European Union without a deal on the future relationship, something some business leaders fear would cause chaos.
Borrowing the Labour leader's slogan "For the many, not the few" she says the Conservatives "are a party not for the few, not even for the many, but for everyone who is willing to work hard and do their best".
"We are all ... members of the Conservative and Unionist party". Northern Ireland is part of that union. "Do not believe that we can somehow get it wrong now and fix it later - get out properly next year, or the year after".
She said: "It has been deeply frustrating to hear people who voted Remain and in Europe talk about Northern Ireland as though we can't touch the Belfast Agreement. Our [Brexit plan] does that, it's the only plan on the table at the moment that does".