In his first words since accepting the result of the EU referendum on Friday, Mr Johnson wrote that "the only change" would be to free the UK from the EU's "extraordinary and opaque" law, which "will not come in any great rush".
'Single market access'
His column said: "I cannot stress too much that Britain is part of Europe, and always will be.
"There will still be intense and intensifying European co-operation and partnership in a huge number of fields: the arts, the sciences, the universities, and on improving the environment.
"EU citizens living in this country will have their rights fully protected, and the same goes for British citizens living in the EU.
"British people will still be able to go and work in the EU; to live; to travel; to study; to buy homes and to settle down. As the German equivalent of the CBI - the BDI - has very sensibly reminded us, there will continue to be free trade, and access to the single market.
"The only change - and it will not come in any great rush - is that the UK will extricate itself from the EU's extraordinary and opaque system of legislation: the vast and growing corpus of law enacted by a European Court of Justice from which there can be no appeal."
In other developments:
Prime Minister David Cameron said on Friday that he would step down as PM by the autumn after losing the vote for Britain to remain within the EU.
Mr Johnson, MP for Uxbridge and the leading pro-Brexit campaigner, is among those tipped to succeed him.
Article 50 of the Lisbon treaty, which sets a two-year deadline on the UK's formal exit from the EU, has not yet been triggered by the British government.
EU foreign ministers have previously urged Britain to start the process soon.
Speaking on Sunday, French President Francois Hollande said there was no going back on the UK's decision, adding: "What was once unthinkable has become irreversible."
On Saturday, German Chancellor Angela Merkel said the EU had "no need to be particularly nasty in any way" in the negotiations with Britain.
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Well done Boris. .. except nobody in Europe like us and have the knives out. There's about 100k jobs in London handling euros that France and German vested interests want. That on its own is irreparable.
From here I hope the knock the pounds got today is not a sign of a market crash as the ftse opens... osbourne to speak next.