Mr Johnson also said the UK would not be “hauling up the drawbridge” even though Prime Minister Theresa May has pledged new migration controls.
Mrs May set out her Brexit strategy - which includes pulling out of the EU single market - in a speech on Tuesday.
She has also warned the EU not to try to “punish” the UK, saying she would walk away unless the right deal was offered.
Mr Johnson denied that Brexit would mean a reduced relationship between the UK and the EU, arguing that Britain’s “joyous exploration” of other European cultures would continue.
Writing in the Daily Telegraph, he said: “We will no longer be part of the common commercial policy, or bound by the Common External Tariff, and we will no longer have our trade policy run by the EU commission.
“That means - crucially - that we will be able to do new free trade deals with countries around the world. They are already queuing up.
“Under EU rules, we are not formally allowed to negotiate these new treaties until we leave. But there is nothing to say that ideas cannot be pencilled in.
“We will continue the joyous exploration of other European culture and civilisation that has been expanding ever since the dawn of cheap air travel, and we will continue to welcome vast numbers of EU tourists to the UK.
“We are not slamming the door to migrants, or hauling up the drawbridge.”