* Turkey blocks deal over next NATO head
* Deadlock threatens unity at 60th anniversary summit
* Obama to push for support on new Afghan policy
* Police fire tear gas as protesters vow to disrupt meeting
(Adds new protests, demonstrator quote)
By Crispian Balmer
STRASBOURG, France, April 4 (Reuters) - Turkey blocked Europe's candidate to head NATO, souring a summit marking the military alliance's 60th anniversary and opening a new rift between Ankara and its Western European allies.
Danish Prime Minister Anders Fogh Rasmussen had emerged as the front-runner to replace the outgoing NATO secretary-general but Turkey objected on Friday, criticising his handling of a row over cartoons of the Prophet Mohammad that offended Muslims.
U.S. President Barack Obama, making his first overseas tour since taking office in January, had signalled to Europe that he would support Rasmussen.
NATO officials put a brave face on the deadlock and said talks would continue on Saturday, the second and last day of the summit that is co-hosted by France and Germany.
"We don't have consensus yet," NATO spokesman James Appathurai told a news conference, adding: "We will get there. This alliance always gets there."
The leaders had hoped to reach a decision on Friday, clearing the way for detailed discussions on Saturday over Afghanistan with Obama promoting his new Afghan strategy and keen for Europe to do more to bolster the war effort.
Hundreds of anti-NATO protesters, who have vowed to disrupt Saturday's meeting after two days of often violent clashes, challenged tight security surrounding the summit before dawn.
A Reuters cameraman said police fired tear gas towards at least one group of protesters on the outskirts of Strasbourg.
"We are going in many different groups so it is less easy to block us," said protesters' spokesman Christoph Kleine.
On Friday, police fired hundreds of rounds of tear gas and rubber bullets in street clashes with protesters.
OBAMA CHARM OFFENSIVE
In sharp contrast, Obama was mobbed by cheering crowds in both France and Germany on Friday, underlining his popularity in Europe which had never warmed to his predecessor George W. Bush.
Obama told an enthusiastic audience of French and German youths that America was changing, but said Europe was more threatened by al Qaeda than the United States because it was closer to the conflict zones.
He said European nations should do more to help in the fight against Islamist militants in the Afghan war, which risks slipping from NATO's control more than 7 years after U.S-backed forces toppled the Taliban from power.
"Europe should not simply expect the United States to shoulder that burden alone," he said, looking to get NATO to back his new Afghan plan which includes sending thousands more troops into the war zone over the coming months.
However, the dispute over the NATO chief risks destroying any pretence of unity in the alliance, which was created soon after World War Two to defend Europe's borders and has continued to expand after the demise of its first foe, the Soviet Union.
German Chancellor Angela Merkel had openly endorsed Rasmussen and sounded confident a deal would be swiftly reached, but she appeared to misjudge Turkey, which argued that the Dane would undermine NATO's reputation in the Muslim world.
"We ask why we got stuck on a single name. Let's look for new alternatives ... This has nothing to do with Rasmussen personally. We just don't want NATO to get harmed," said Turkish Prime Minister Tayyip Erdogan.
The dispute could have serious ramifications for Ankara's tortuous bid to join the European Union.
Negotiations over EU membership are blocked at a number of levels and neither Germany nor France, the traditional engines driving the union, are likely to promote the talks if Ankara persists in blocking their favoured NATO candidate. (Additional reporting by David Brunnstrom, Matt Spetalnick, Caren Bohan, Gilbert Reilhac, Sophie Hardach; Writing by Crispian Balmer, Tim Hepher; Editing by Ralph Gowling)