His demand came at a parliamentary debate on the Brexit result on Tuesday.
To loud applause from other members, Juncker said, "I am sad about this result and make no secret of that but I am calling on the UK to clarify its position as soon as possible. We cannot be embroiled in any lasting uncertainty."
He said he had instructed all European Commissioners - all of whom were sat with him in the plenary - and DGs in the Commission not to "engage in secret negotiations" with British officials over the terms of its EU exit.
Ukip leader Nigel Farage took his place along with other MEPs in the Brussels chamber as Juncker addressed members about the next steps.
"That's the last time you are applauding here," Juncker said, after the fiercely anti-EU politician applauded his opening statement that Europe, "must respect British democracy and the way it has expressed its view."
"To some extent I am really surprised that you are here," he told him.
"You were fighting for the exit, the British people voted in favour of the exit. Why are you here?"
Juncker also took a swipe at press speculation that he was beset by ill health, saying, "I am not tired and I am not sick, contrary to reports in the German press. Apparently, journalists are now doctors but, be assured, to my last breath I will fight for a united Europe."
"The UK clearly wants to leave the EU so we must now act accordingly," he told the packed plenary.
He added, "We have lost something very important and one of our wings has been cut off but the EU dream will continue."
In the debate, Farage came under ferocious attack from leading MEPs.
He was accused of "lying" to the British public over promises about spending and migration during the Brexit campaign.
ALDE group leader Guy Verhofstadt likened the infamous pre-referendum poster Farage had promoted showing lines of migrants supposedly queuing to enter the UK to "Nazi propaganda" and accused him and other people in the Leave camp of creating a "climate of fear and negativity."
The Belgian deputy said, "This was the most shocking thing about the whole debate."
The attack on Farage was also taken up by EPP group leader Manfred Weber, who also accused him of lying, adding, "If he had an ounce of decency he would apologise today to the British public."
In a highly charged debate, which comes as EU leaders convene in Brussels for a key two-day summit, the German MEP shouted to Farage: "Shame on you."
Weber said that, as 73 per cent of young Britons had voted to stay in the EU, "the futures of a whole generation of young Britons have been destroyed."
Many of those taking part in the hastily convened debate agreed that Britain must now act quickly to start its EU exit negotiations.
Further comment came from S&D group leader Gianni Pittella who said the UK had to quickly launch exit talks in order to "stop paralysis in the EU."
He added, "We cannot be held hostage to internal problems in the UK."
The Italian said that with a falling pound and talk of another independence vote in Scotland, the result had left a "battleground of destruction."
However, ECR group leader Syed Kamall, who supported Brexit, said that while there was "shock on both sides of the channel", now was not the time for snap decisions.
He added, "It is not the speed at which this is done but the deal we get at the end of it that is important."
In a sometimes stormy debate, Jeanine Hennis-Plasschaert, representing the Dutch EU Council presidency, said the risk of the prolonged uncertainty was that the EU would be left in "limbo."
She added, "The ball is now in the UK's court but we must give Britain time to take all the necessary decisions. Now is the time for cool heads."
Hennis-Plasschaert, who is the Dutch defence minister, described the result as a "wake up call" for the EU, adding, "The concerns clearly shown here are shared by many others in Europe. The message is that we have to do better."
European Council President Donald Tusk was absent from the chamber, but Parliament's President Martin Schulz led glowing tributes to the UK's outgoing Commissioner, Jonathan Hill, who announced he will now leave the Commission.
Hill, who campaigned for the Remain side, received a standing ovation from other Commissioners and most MEPs.
Schulz, who said it was the first time Parliament had ever convened such a session, said Hill had been put in a "very difficult position" after the result.