Johnson is favoured to succeed Prime Minister David Cameron, who said he would resign before autumn.
"The British people have voted to leave the European Union and their will must be respected," Cameron said in a press conference Friday morning.
"I will do everything I can to steady the ship over the coming weeks, but I do not think it would be right for me to try to be the captain that steers our country to its next destination."
The ex-London mayor, surrounded by police, was booed by crowds outside his home as he got into a cab headed to Vote Leave headquarters.
Gareth Fuller / PA Wire/Press Association Images Boris Johnson leaves his home in north London, after David Cameron announced he will quit as Prime Minister by October.
Britain voted to leave the European Union by a final vote of 51.9% to 48.1%.
The historic outcome has sent shockwaves through the markets. The pound collapsed to a 30-year-low after several TV news stations called Leave as the winner in the early hours Friday morning in London.
The Bank of England Governor Mark Carney tried to calm fears in a statement Friday morning. "The capital requirements of our largest banks are now 10 times higher than before the financial crisis," he said, adding that "The Bank of England has stress-tested those banks against scenarios far more severe than our country currently faces," he said. Carney added said the Bank is "ready to provide more £250 billion of additional capital to its normal operations."
UK Independence Party leader Nigel Farage told ITV's "Good Morning Britain" that he didn't know which Conservative politician he would like to step up to be prime minister, but that he doesn't want the job.