* Say Qatar flouts pledge not to meddle in others' affairs
* Withdrawal of ambassadors unprecedented in GCC history
* Qatar says "regrets" move but will not retaliate
* Qatar is GCC maverick, backing Muslim Brotherhood
* Gulf Arab allies split over upheaval in Syria, Egypt (Egypt denies extra airport security, ambassador stays in Cairo)
By Angus McDowall and Amena Bakr
RIYADH/DOHA, March 5 (Reuters) - Saudi Arabia, the United Arab Emirates and Bahrain withdrew their ambassadors from Qatar on Wednesday in an unprecedented public split between Gulf Arab allies who have fallen out over the role of Islamists in a region in turmoil.
Qatar's cabinet voiced "regret and surprise" at the decision by the fellow-members of the six-nation Gulf Cooperation Council, but said Doha would not pull out its own envoys and that it remained committed to GCC security and stability.
The Saudi-led trio said they had acted because Qatar failed to honour a GCC agreement signed on Nov. 23 not to back "anyone threatening the security and stability of the GCC whether as groups or individuals - via direct security work or through political influence, and not to support hostile media".
Saudi Arabia and the UAE are fuming especially over Qatar's support for the Muslim Brotherhood, an Islamist movement whose political ideology challenges the principle of dynastic rule.
They also resent the way Doha has sheltered influential Brotherhood cleric Yusuf Qaradawi and given him regular airtime on its pan-Arab satellite television channel Al Jazeera.
The GCC, which normally keeps its disputes under wraps, is a pro-Western alliance of monarchies set up in the 1980s to counter Iranian influence in the Gulf, and includes several of the world's biggest producers and exporters of oil and gas.
Kuwait and Oman did not join the diplomatic rebuke to Qatar. Kuwait's parliament speaker Marzouq al-Ghanim said he was concerned by its implications. Oman has not commented.
Saudi Arabia, the biggest GCC state by population, size and economy, has grown increasingly frustrated in recent years by the efforts of Qatar, a country of just 2 million, to leverage its large wealth from gas exports into regional clout.
Qatar's stock market tumbled 2.3 percent after Wednesday's announcement. There is significant cross-border investment in the stock markets of GCC countries by investors from other GCC nations. Saudi investors play a major role in all GCC markets.
Saudi Arabia has tried for two years to align the foreign and security policies of Sunni-ruled GCC states to combat what it sees aggression by Shi'ite Iran, its regional arch-rival.
Gulf analysts and diplomats say it is too early to say if the rift with Qatar will break the GCC apart, pointing to previous rows between member states that were later settled.
CHALLENGE FOR RULER
Qatar has been a maverick in the conservative Gulf region, backing Islamist movements in Egypt, Syria and elsewhere that are viewed with suspicion or hostility by some GCC members.
The latest ruckus is a challenge for Qatar's youthful new ruler, Sheikh Tamim bin Hamad al-Thani, who suggested when he succeeded his father in June that he would pursue Doha's assertive, independent-minded foreign policy.
Saudi and other Gulf Arab officials, as well as Egypt's military-backed rulers often complain about Al Jazeera, which they see as openly pro-Brotherhood and critical of their own governments. Al Jazeera says it is an independent news service giving a voice to everyone in the Middle East.
Three Al Jazeera journalists are in jail in Egypt, accused of helping a terrorist organisation - code for the Brotherhood.
An Egyptian Foreign Ministry spokesman, Badr Abdelatty, denied an assertion by Egyptian airport sources cited by Reuters that Qatari citizens would be subject to extra security screening measures to make sure they were not involved in "hostile acts" against Egypt. He said the report was nonsense.
But Abdelatty expressed understanding for the withdrawal of ambassadors, saying Qatar had to move away from policies and positions that fragmented Arab unity.
He said the Egyptian ambassador to Qatar had been back in Cairo since early February. "Our decision to keep him here is a political decision. We have specific demands...not to interfere in our internal affairs," he said.
Saudi Arabia, the UAE and Bahrain said they had acted after GCC foreign ministers meeting in Riyadh on Tuesday had failed to persuade Qatar to implement the Nov. 23 accord.
Qatar suggested the move stemmed from displeasure over its actions beyond the Gulf, for example in Syria and Egypt, where it has backed groups opposed to the Saudi government.
It said the action "had nothing to do with the interests, security and stability of GCC peoples but rather a difference in positions on issues external to the Gulf Cooperation Council".
Qatar, which also backed Libyan rebels who toppled Muammar Gaddafi, says it supports Arab people against oppression.
A source close to the Saudi government said pressure on Qatar would continue until it changed its policies. "They have to divert their position on many issues and we are waiting for real signs of this, not just talk." (Reporting by Angus McDowall in Riyadh, Reem Shamseddine in al-Khobar, Rania el Gamal in Dubai, Amena Bakr in Doha, Sylvia Westall in Kuwait and Maggie Fick in Cairo; Writing by William Maclean; Editing by Alistair Lyon)