The former London mayor said he was "sad" David Cameron had decided to stand down
Boris Johnson has said Britain's vote to leave the EU is a "glorious opportunity" for it to find its "voice in the world again".
The former London mayor said the European Union was "a noble idea for its time", but it was "no longer right for this country".
He said: "I believe we now have a glorious opportunity - we can pass our laws and set our taxes entirely to the needs of the UK economy
"We can control our 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.
"Above all, we can find our voice in the world again... powerful, liberal, humane, an extraordinary force for good in the world."
Directly addressing the millions of young voters who voted to stay in the EU, Mr Johnson insisted the result did not represent a retreat into isolationism and that Britain would remain a "great European power".
He said: "We cannot turn our backs on Europe. We are part of Europe. Our children and grandchildren will continue to have a wonderful future as Europeans travelling to the continent, understanding the languages and cultures, that make up of common European civilisation."
He added: "It is the essence of our case that young people in this country can look forward to a more secure and more prosperous future if we take back the democratic control that is the basis of our economic prosperity."
Mr Johnson, who campaigned against prime minister David Cameron, paid tribute to his Conservative party leader.
He said he was "sad" Mr Cameron had decided to stand down, "but I respect his decision".
Mr Johnson said the PM's "bravery" gave the nation its first referendum on the EU, and he described Mr Cameron as "one of the most extraordinary politicians of our age".
Fellow Conservative and Vote Leave campaigner Michael Gove, a friend of the PM, also paid tribute to Mr Cameron, saying he deserved to be remembered as a "great prime minister".
He promised the British public that Brexit would be implemented "calmly, co-operatively and consensually", declaring the UK is "open for business, open to trade, open to other cultures, open to the world".
Both Mr Johnson and Mr Gove were among more than 80 Tory MPs who had earlier signed a letter saying the Prime Minister had a "mandate and a duty" to stay on whatever the result.
Neither politicians revealed whether they would be putting their names forward to replace Mr Cameron.