Now touted as Mr Cameron's most likely successor, Mr Johnson called his party leader a "principled man" who was "one of the most extraordinary politicians of our age".
He also praised his senior's "superb leadership" of the Conservative Party, and his "bravery" for supporting the referendum vote.
"Some people are now saying that was wrong and that the people should never have been asked in this way. I disagree," he said at Vote Leave's headquarters in London, flanked by Labour MP and Brexit campaigner Gisela Stuart, and fellow Conservative Michael Gove.
"I believe it was entirely right and inevitable and indeed that there is no way with dealing with a decision on this scale except by putting it to the people.
"In the end, this question is about the people, it’s about the right of the people of this country to settle their own destiny, it’s about the very principles of our democracy.
"The rights of all of us to elect and remove the people who make the key decisions in their lives. And I think that the electorate have searched in their hearts and answered as honestly as they can."
The Uxbridge and South Ruislip MP said the Brexit vote "does not mean that the United Kingdom will be in any way less united" or "less European".
"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 said.
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.
“We cannot turn our backs on Europe, we are part of Europe,” he said.
“Our children and our grandchildren will continue to have a wonderful future as Europeans, travelling to the continent, understanding the languages and cultures that make up our common European civilisation.”
The former Mayor of London also said change would not be immediate.
"There is no need for haste," said Mr Johnson.
"As the Prime Minister has just said, nothing will change over the short term except that work will have to begin on how to give effect to the will of the people and to extricate this country from the supranational system.
"To those who may be anxious, whether at home or abroad, this does not mean that the United Kingdom will be in any way less united, nor indeed does it mean that it will be any less European.
“I believe we now have a glorious opportunity. We can pass our laws and set our taxes entirely according to the needs of the UK economy.
"We can control our own borders in a way that is not discriminatory but fair and balanced and take the wind out of the sails of the extremists and those who would play politics with immigration.
"I want to reassure everybody that, in my view, as a result of this Britain will continue to be a great European power, leading discussions on defence and intelligence sharing and all the work that currently goes on to make our world safer.
“But there is simply no need in the 21st century to be part of a federal system of government, based in Brussels that is imitated nowhere else on Earth. It was a noble idea for its time, it is no longer right for this country.”
He also appealed directly to "young people" not to feel dismayed by the vote, but to know that great opportunities lay ahead for them.
And he said the debate must move on from those who "would play politics with immigration".
Mr Johnson spoke shortly after Mr Cameron formally told the Queen of his decision to hand over the premiership to a new Conservative leader by October.
More than £100 billion was wiped off the FTSE 100 as the index fell more than 7 per cent, while the pound also crashed 8 per cent against the US dollar.
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