The EU constitution
BBC European affairs correspondent William Horsley says Mr Juncker seemed so distressed that he could hardly take in the fact of the second "No" vote.
The mood in Brussels is deep gloom, our correspondent says.
Latvia's parliament ratified the constitution by a vote of 71-5 on Thursday, bringing the number of countries backing it to 10.
The charter needs to be approved by all 25 of the EU's member states by the end of October next year in order to become law.
'Cheeks slapped'
Some EU officials tried to put a brave face on the Dutch result, but others were more plain-spoken.
"The French slapped the left cheek of Europe, the Dutch have now slapped the right," Graham Watson, Liberal leader in the EU Parliament, told Reuters news agency.
"I hope this will bring the European Union out of its torpor and force its leaders... to show leadership."
Provisional results from the Netherlands show nearly two-thirds of voters were against the draft treaty. Turnout was 62.8%.
The euro fell to an eight-month low against the dollar as the Dutch results came out.
European Commission head Jose Manuel Barroso is meeting European Parliament leaders on Tuesday for talks in the wake of the referendums.
"We have a serious problem, but we must continue our work," Mr Barroso said.
Mr Juncker said other EU members should go ahead with their referendums.
'Profound questions'
UK Foreign Secretary Jack Straw said the double rejection "now raises profound questions for all of us about the future direction of Europe".
Europe has spoken loud and clear - it is time for change 
Nicosia, Cyprus
Britain will take over the rotating presidency of the European Union next month.
Dutch Prime Minister Jan Peter Balkenende said he was "very disappointed" with his country's referendum but pledged to honour the voters' verdict, which is not legally binding.
Neelie Kroes, a European commissioner from the Netherlands, told the BBC her country's "No" vote was partly a reaction to the speed at which the union was moving.
"What the 'No' is about in the Netherlands is that it's going too quick. Sometimes people have the feeling that in the EU they are interfering in too many national affairs and are doing too much."
Wednesday's ballot was the first time Dutch voters have had a chance to vote directly on a European Union issue - and the first national referendum in two centuries.
With some postal ballots still to be counted, the official result is not expected until 6 June.
The draft EU constitution was signed last year, after lengthy negotiations between members states.
It brings together for the first time the many treaties and agreements on which the EU is based. It defines the powers of the EU, stating where it can and cannot act and where the member states retain their right of veto.
It also defines the role of the EU institutions.
The constitution was rejected by 55% of French voters in a referendum on Sunday.