* Some Arab states willing to police the region themselves
* Raids seen as message to Qatar to stop backing Islamists
* Arab states increasingly active in each others' crises
* UAE and Egypt deny involvement, U.S. backs off statement (Adds State Department statement)
By William Maclean
DUBAI, Aug 26 (Reuters) - Air strikes against Libyan Islamist militants that U.S. officials had said were staged by Egypt and the United Arab Emirates could mark an escalation of a regional struggle over the future of the Arab world.
Arab responsibility for the attacks would add to a picture of the West's regional allies acting increasingly independently in the absence of decisive U.S. involvement, seeking security goals with which Washington might not agree.
Egypt publicly denied involvement in the attacks on targets in its western neighbour and sensitivity over the strikes led to confusing statements from the administration in Washington.
U.S. officials told Reuters on condition of anonymity on Monday that the UAE had flown a series of strike missions from Egypt, and State Department and Pentagon spokespeople on Tuesday initially affirmed their understanding that Egypt and UAE were involved.
Later in the day the State Department backed off comments on Libya made at a regular briefing, saying they were "intended to refer to countries reportedly involved, not speak for them."
In a joint statement on Monday, the United States and its European partners Britain, Germany, Italy and France had urged outsiders not to interfere in Libya, which is suffering its worst violence since the overthrow of Muammar Gaddafi in 2011.
Tripoli residents said last weekend that unidentified jets had attacked targets in the capital. There were also strikes on Islamist-held positions last Monday.
Egypt formally denied conducting the air raids and UAE Minister of State for Foreign Affairs Anwar Gargash suggested on Twitter that the allegations had been promoted by anti-UAE Islamists.
Whoever carried out the raids, they were in tune with wider efforts by Egypt and conservative Sunni Muslim allies to roll back the influence of the Muslim Brotherhood - a regional Islamist movement - and its sponsor, Qatar.
Analysts noted that U.S. President Barack Obama, who last year called off air strikes on Syria at the last minute, has himself said allies in the region should play a greater role in tackling local crises.
"In the light of U.S. inaction in Syria, the message is clear, that you have to take care of your own concerns," said Emirati political scientist Abdulkhaleq Abdulla, stressing that he did not know for sure if the UAE was involved or not.
If the raids were indeed carried out by Egypt and the UAE, it would open a new chapter in inter-Arab relations, said Theodore Karasik, research director at Dubai think tank INEGMA.
"The feeling is that America hasn't stood up for its values and policies in the region," he said, referring to a common Arab view that the U.S. administration has been hesitant in supporting rebels fighting Syrian President Bashar al-Assad.
"So these states will now take it upon themselves to act. Ironically, this is, in broad terms, what Washington has been asking them to do - solve their own problems."
The alleged use of outside military muscle touched a nerve in the West, acutely aware that its own intervention in Libya in the run-up to the fall of Gaddafi contributed to the country's descent into chaos.
In an indication of the sensitivity of the issue, the publication of their assertions was followed within hours by a joint statement by the United States and European allies cautioning against foreign interference.
Outside involvement would worsen divisions in Libya and slow progress in its political transition, it said.
And yet the West may have to get used to a more activist stance by participants in a tussle for influence pitting Egypt and most of the conservative Gulf Arab states against Islamist-friendly Qatar, Sudan and non-Arab Turkey and Iran.
A number of Arab powers have used a variety of tools in the past four years, including armed force, aid, finance and diplomacy to shape events in Egypt, Yemen, Syria and Libya to their advantage.
"The important point here is that regional forces are taking their own path to supporting proxies," said Karasik. "This is the result of the region wanting to police itself without waiting for extra-regional decisions."
Abdulla said that, if the UAE had taken part in the raid, it must have had "very compelling reasons to do so." If Libya became a failed state and an exporter of extremists then the stability of neighbouring Egypt would be at risk, he added.
The world was busy with many other crises, and so action might have been needed to prevent extremists from taking over, he said.
While policy differences between Washington and its Arab allies are nothing new, the propensity of some to go it alone in pursuing their aims is novel.
Egypt provides a clear example. Saudi Arabia was furious when veteran ruler Hosni Mubarak was toppled in 2011 and the Muslim Brotherhood, long mistrusted by Riyadh, later won power. Qatar helped to fund the elected Brotherhood government of President Mohamed Mursi, who was ousted by the army last year.
Riyadh and the UAE have since provided money to support Abdel Fattah al-Sisi, who as a general led the takeover and has since been elected president after suppressing the Brotherhood.
U.S. officials have looked askance at Sisi's heavy-handed political and security tactics, which they believe have helped to polarise Egyptian society.
Sisi's men now fear that Islamists, if left to flourish in Libya's disorder, could lay the foundations for the return one day of the Brotherhood in Egypt.
For most Gulf Arabs, the Brotherhood is anathema because its ideology challenges the principle of conservative dynastic rule long followed in the Gulf.
Gulf Arab states take Egypt's stability seriously, regarding the Arab world's most populous nation as their chief regional ally in their confrontation with Shi'ite Muslim Iran.
Riyadh sees the Iranian administration as an expansionist power bent on exporting revolution to the Arab world and interfering in the affairs of neighbouring Gulf states. Tehran denies any such interference. (Additional reporting by Lesley Wroughton in Washington. Editing by David Storey and Nick Macfie)