Brexit voters must "build bridges" with Remain supporters who feel "loss and confusion", Boris Johnson says.
The leading pro-Leave campaigner and Tory leader frontrunner said the 52-48 result was "not entirely overwhelming".
Setting out his post referendum vision in the Daily Telegraph, he said the UK could reform its immigration system and keep access to the single market.
But ex-chancellor Alistair Darling accused him of treating the situation "like a big game".
Thursday's EU referendum, which has triggered a Conservative leadership contest and mass resignations from Labour's front bench, followed a bitter four-month campaign with recriminations continuing over the weekend.
"We who are part of this narrow majority must do everything we can to reassure the Remainers," wrote Mr Johnson, who has kept a low profile since the result was announced.
"We must reach out, we must heal, we must build bridges - because it is clear that some have feelings of dismay, and of loss, and confusion."
He denied that the Leave vote had been mainly driven by immigration, saying restoring control of British democracy was "the main issue".
He said there would still be "intense" cooperation between the UK and the EU on arts, science and the environment and said Britons would still be able to travel and work in Europe.
In other developments:
- UK shares have remained volatile in the wake of the Brexit vote, while the pound fell further against the dollar
- Prime Minister David Cameron chaired the first meeting of the cabinet since the EU referendum result. It is not a political cabinet and Mr Johnson will not be there
- Mr Cameron is also due to address MPs later
- The executive of the 1922 Committee of backbench Tory MPs is set to meet to draw up the timetable for the Tory leadership contest
- German Chancellor Angela Merkel and French President Francois Hollande will hold talks later in Berlin to discuss the fallout of Brexit
Mr Johnson said Britain would always be "part of Europe" and "there will continue to be free trade, and access to the single market".
Speaking outside his home on Monday morning, he said the status of EU nationals living in the UK and Britons abroad would be protected under what he called a "fair, impartial and humane" immigration system.
He said "project fear" - the term he has repeatedly used to attack the Remain campaign - was "over" after Chancellor George Osborne indicated there would be no immediate emergency Budget.
Mr Osborne, who warned during the campaign of spending cuts and tax rises if there was a vote to leave, said there would still need to be an "adjustment" in the UK economy, but said it "perfectly sensible to wait for a new prime minister" before taking any such action.
Europe's Brexit crisis - the week ahead
- Monday - German Chancellor Angela Merkel holds crisis talks in Berlin, first with European Council President Donald Tusk, then with French President Francois Hollande and Italian Prime Minister Matteo Renzi (statement to media expected at 18:30 local time (16:30 GMT), then more talks over dinner)
- Tuesday - Extraordinary European Parliament session in Brussels on Brexit vote 10:00-12:00 (08:00-10:00 GMT), including speeches by Mr Tusk, European Commission President Jean-Claude Juncker, and party leaders, probably including UKIP's Nigel Farage;
- Later Tuesday - EU summit (European Council) in Brussels: Prime Minister David Cameron to brief the other EU leaders over dinner, from 19:45 (17:45 GMT), explaining the political fallout in the UK
- Wednesday - Second day of EU summit: Breakfast talks between 27 leaders - Mr Cameron not attending. Talks focus on UK's "divorce process" as stipulated by EU's Article 50 and Mr Tusk will "launch a wider reflection on the future of the EU"; press conferences in afternoon
Speaking for the first time since Leave's victory, the chancellor said only the UK could trigger Article 50 of the Lisbon treaty, which sets a two-year deadline on the UK's formal exit from the EU.
Prime Minister David Cameron, who will step down by the autumn, has said responsibility for triggering Article 50 would fall to his successor.
Another minister touted as a possible leadership contender, Work and Pensions Secretary Stephen Crabb, said whoever is elected leader would have to work with Scottish First Minister Nicola Sturgeon and have a "plan for holding this United Kingdom together".
Ms Sturgeon says a second independence referendum is "highly likely" after Scotland voted to Remain in the EU.
Speaking on BBC Radio 4's Today programme, Mr Darling, who campaigned for Remain, said he was more worried about the UK economy now than in 2008 during the financial crisis because there were "so many uncertainties, so many unknowns".
He said there did not seem to be a plan for what to do, and warned of a "vacuum" between now and October, when the next Conservative leader is expected to be appointed.
Mr Darling said: "We got no government, we have got no opposition, the people who got us into this mess have run away - they have gone to ground."
But Commons leader and Leave campaigner Chris Grayling said preparations would be needed for the coming months.
"We've clearly got to take things forward, we cannot sit on our hands for the next four months, that goes without saying and we'll be setting out more of that in due course," he said.
Asked whether the UK could concede some freedom of movement in exchange for access to the single market, he said last week's vote was a clear mandate for the government to put controls over immigration, and that it "had to happen".
EU politicians have urged Britain to start the Article 50 process soon.
Michael Fuchs, who is vice chair of German Chancellor Angela's Merkel's CDU party, said it would not be possible for the UK to retain access to the single market without free movement.
"Either you are in a club or you are out of a club," he told Today.
"If you are in a club you have to follow the rules. If you are out of the club, there will be different rules."