Anders Fogh Rasmussen had been the odds-on favourite
Nato leaders have failed to agree on a new head for the alliance on the first day of their summit in Germany.
Big European powers had backed Danish PM Anders Fogh Rasmussen as secretary general and Germany's Angela Merkel urged leaders to endorse him on Friday.
But Turkey has raised objections - notably over Mr Rasmussen's handling of controversial Danish cartoons depicting the Prophet Muhammad published in 2005.
The two-day summit, co-hosted by France, will continue on Saturday.
The meeting, marking the 60th anniversary of the alliance, is dominated by Nato's mission in Afghanistan.
BBC diplomatic correspondent Jonathan Marcus, who is at the summit, says the failure to agree quickly on a new secretary general is an embarrassment - but leaders do not want it to overshadow their work.
OBAMA'S TRANSATLANTIC VISIT
3 April: Obama meets Sarkozy in France and Merkel in Germany
4 April: Leaders walk across the Rhine and hold North Atlantic Council meeting in Strasbourg
5 April: Obama in Prague for US-EU summit
6-7 April: Obama visits Turkey
On Friday, US President Barack Obama called for a more effecitive use of Nato resources in the country and underlined the threat posed by militants.
Speaking after talks in Strasbourg with French President Nicolas Sarkozy, Mr Obama said: "It is probably more likely that al-Qaeda would be able to launch a serious terrorist attack in Europe than in the United States because of proximity.
"This is not an American mission, this is a Nato mission, this is an international mission."
UK Prime Minister Gordon Brown, for his part, offered a temporary increase in British troops serving in Afghanistan.
The UK currently has 8,300 soldiers in the country, the second-largest international contingent of troops after the US.
On the issue of wider defence, Mr Obama said: "We would like to see Europe have much more robust capabilities."
NATO: KEY MOMENTS
Founded 1949, largely to block Soviet expansion into Europe
Twenty-six member states who vow to defend each other
Militarily dominated by the US
Acted in non-member state for first time in 1995 - implementing military aspects of Bosnia peace accord
Operated outside Europe for first time in 2003 - in Afghanistan
He added: "We're looking to be partners with Europe, and the more capable they are defensively, the more we can act in concert on the shared challenges we face."
Later, Mr Obama told an audience in Strasbourg that the US and Europe had allowed the alliance to drift in recent years.
He said the US had been "arrogant" and "dismissive" towards its allies, while there was "insidious" anti-Americanism in Europe. He said these attitudes had to change.
"I've come to Europe this week to renew our partnership - one in which America listens and learns from our friends and allies. But where our friends and allies bear their share of the burden," he said.
Mr Obama also announced a long-term goal of "a world without nuclear weapons", and said he would outline details of his non-proliferation proposal in a speech in Prague on Sunday.
"Even with the Cold War now over, the spread of nuclear weapons or the theft of nuclear material could lead to the extermination of any city on the planet," he said.
Mr Obama went on to Baden-Baden in Germany, where he met Chancellor Angela Merkel.
"Leadership from our two countries is going to be critical" in meeting Nato's future challenges, he said at a joint news conference with Mrs Merkel.
The German towns of Kehl and Baden-Baden are both hosting Nato events for the alliance's 60th-anniversary summit.
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At a working dinner in Baden-Baden on Friday, leaders had been expected to agree on a successor to Secretary General Jaap de Hoop Scheffer, who steps down at the end of July.
Mr Fogh Rasmussen has been seen as the frontrunner to succeed him, and won strong endorsement from Mrs Merkel and several other leaders.
But Turkey expressed opposition, based on Mr Rasmussen's stance over the publication of the caricatures of the Prophet Muhammad by a Danish newspaper.
Many Muslims around the world were angered when Mr Fogh Rasmussen refused to censure the paper, invoking freedom of speech.
Turkey also objects to a Kurdish TV station broadcasting from Denmark.
Speaking in London shortly before the summit, Turkish Prime Minister Recep Tayyip Erdogan said: "I look at [his candidacy] negatively."
Talks are due to resume in Strasbourg on Saturday. Nato leaders are also expected to consider relations with Russia.