EU leaders have insisted that the UK must move swiftly to negotiate leaving the organisation, saying any delay would prolong uncertainty.
European Commission head Jean-Claude Juncker stressed the “Union of the remaining 27 members will continue”.
The UK voted by 52% to 48% to leave the EU, and David Cameron has announced he will step down as PM by October.
He has said it will be up to the new PM to invoke the article that will begin the UK’s withdrawal.
Global stock markets fell heavily on the news of the so-called “Brexit”. The value of the pound has also fallen dramatically.
A meeting of 27 EU leaders has been scheduled for Wednesday to discuss Brexit – but Mr Cameron is not invited.
Mr Juncker held crisis talks with European Parliament President Martin Schulz, president of the European Council Donald Tusk and Dutch PM Mark Rutte on Friday morning.
They then released a statement saying they regretted but respected the British decision.
They called for the UK “to give effect to this decision of the British people as soon as possible, however painful that process may be. Any delay would unnecessarily prolong uncertainty”.
They added that the deal agreed with Mr Cameron in February to protect London’s financial markets, curb immigration and opt out of closer union “ceases to exist” and “there will be no renegotiation”.