Mr Johnson said the Brexit vote gave the UK a “glorious opportunity” for a brighter future, and insisted that Britain would remain “a greater European power” outside the future Union of 27 nations.
He was speaking shortly after Mr Cameron visited Buckingham Palace to formally inform the Queen of his decision to hand over the premiership to a new Conservative leader by October.
The Prime Minister announced his decision outside 10 Downing Street hours after Leave secured a shock victory in the historic referendum, and minutes after the markets opened with carnage in the City of London.
More than £100 billion was wiped off the Ftse 100 as the index fell more than 7%, while the pound also crashed 8% against the US dollar.
But the markets regained some of the lost value after Bank of England governor Mark Carney pledged to intervene to shore up the markets.
Flanked by wife Samantha as he delivered an emotional statement outside Number 10, Mr Cameron said he accepted the decision of the electorate, which voted by 52% to 48% to quit the EU.
“The British people have voted to leave the European Union and their will must be respected,” he said. “The will of the British people is an instruction that must be delivered.”
He said that while he would do everything he could to “steady the ship” over the coming weeks and months, he was not the right person to be “the captain that steers our country to its next destination” and would hand over to a new leader by the time of the Conservative Party conference.
His announcement will trigger a battle for the Conservative leadership — and the keys to Number 10 — likely to feature Brexit standard-bearer Mr Johnson taking on figures such as Home Secretary Theresa May, who kept a low profile in the referendum campaign.
Mr Cameron said he would leave it to his successor to invoke Article 50 of the Lisbon Treaty, which kicks off the two-year process of negotiating a new trade relationship with the UK’s former partners. In a press conference at Vote Leave headquarters in London, Mr Johnson said there was no need for “haste” or for invoking the Article, but said work should begin to “extricate this country from the supernational system”.
The former London mayor said: “I believe the British people have spoken up for democracy in Britain and across Europe and I think we can be very proud of the result.”
He insisted that the Brexit vote “doesn’t mean that the UK will be in any way less united, nor indeed does it mean we will be any less European”.
Senior EU representatives said the UK should implement Brexit “as soon as possible, however painful that process may be”, warning that “any delay would unnecessarily prolong uncertainty”.
Mr Cameron has summoned the Cabinet to meet on Monday, the day before he goes to Brussels for a summit where he will “explain the decision the British people have taken and my own decision” to leaders of the remaining 27 member states.
Announcing his resignation after six years as PM — and just 13 months after securing an absolute majority for the first time — was “not a decision I have taken lightly”, said Mr Cameron.
But he said it was “in the national interest to have a period of stability and then new leadership”.
A clearly emotional Mr Cameron said he stood by his assertion that the UK could “find a way” to survive outside the EU.
“Now the decision has been made to leave we need to find the best way and I will do everything I can to help,” he said.
Minutes after the PM’s statement, Mr Carney announced he was making £250 billion available to support markets as he pledged that the Bank of England “will not hesitate to take additional measures as required as markets adjust and the UK economy moves forward”.
Mr Cameron’s defeat was a humiliating turnaround which he can never have imagined when he first announced plans for a referendum to see off the electoral threat of Ukip in 2013 or when he named the date following a renegotiation of Britain’s membership earlier this year.
Even as polling stations closed on Thursday night, most observers, pollsters, and bookmakers were expecting victory for Remain, albeit by the narrowest of margins.
Ukip leader Nigel Farage even came close to conceding defeat, admitting he believed that Remain had “nicked it” and vowing to fight on for withdrawal from the EU, while senior pro-Brexit Tories including Mr Johnson and Michael Gove signed a letter calling on the PM to stay on regardless of the result.
But emphatic early victories in cities such as Sunderland and Swindon made clear the momentum was all on the Brexit side. As dawn arrived, there was no doubt that Leave had secured enough support in the English shires and former Labour strongholds in the North, Wales, and Midlands to sweep aside strong Remain performances in London, Scotland, and Northern Ireland.