Mr Johnson said that means the UK must "come together and move forward united, unleashing the enormous potential of the British people."
Mr Johnson, who is spending New Year with his girlfriend Carrie Symonds on the private island of Mustique in the Caribbean, said he is determined to be a "prime minister for everyone", including the millions who backed Remain or did not vote Conservative in the general election.
"If you are one of them, I want to reassure you that I will be a prime minister for everyone, not just those who voted for me," he said.
In an attempt to quell concerns about the future of the NHS, the prime minister said leaving the EU would enable the government finally to focus on the "people's priorities".
"The loudest message I heard during the election campaign is that people expect us - expect me - to protect and improve the NHS," he said.
"The NHS is a wonderful British invention, there for us and our families when we are ill, whatever our background and regardless of ability to pay. So the NHS will always be my top priority."
The prime minister said that he was also determined to make the UK "the best place on Earth" for "quality education and cutting-edge science".
"Our vision is clear: to unite and level up across the whole United Kingdom - spreading opportunity more fairly - with better infrastructure, superb education and high technology," he said.