* Kerry says Iraqi Sunnis, Kurds and some Shi'ites unhappy
* Iraqi leadership must be inclusive, share power
* Will discuss impact of any oil supply disruption with Gulf countries
* Cairo visit a day after court confirms mass death sentence (Recasts with Iraq comment)
By Lesley Wroughton
CAIRO, June 22 (Reuters) - Secretary of State John Kerry said on Sunday the United States wanted Iraqis to find an inclusive leadership to contain a sweeping Islamist insurgency but Washington would not pick or choose who rules in Baghdad.
Kerry was speaking at the start of a Middle East tour after talks in Cairo with Egypt's new President Abdel Fattah al-Sisi which covered Western concerns over Egypt's crackdown on the Muslim Brotherhood and the fallout of the crisis in Iraq.
Governments across the Middle East and beyond have been alarmed at the speed with which Sunni militants drove Shi'ite Prime Minister Nuri al-Maliki's troops out of large parts of north and west Iraq and pushed towards the capital.
The United States is sending 300 military advisers to Iraq, prompting Iran's Supreme Leader Ayatollah Ali Khamenei to accuse Washington of trying to regain control of the country it once occupied and rule "by its stooges" - a charge Kerry denied.
"The United States is not engaged in picking or choosing or advocating for any one individual, or series of individuals, to assume the leadership of Iraq," he said. "That is up to the Iraqi people and we have made that clear since day one."
However he said the U.S. noted dissatisfaction with Maliki's leadership by Kurds, Sunnis and some Shi'ites and wanted Iraqis to find a leadership "prepared to be inclusive and share power".
At a joint news conference with Egyptian Foreign Minister Sameh Shukri, Kerry also defended U.S. involvement in the Middle East when asked whether U.S policy had caused recent violence in Iraq and Libya.
"What is happening in Iraq is not happening because of the United States in terms of the current crisis. The United States shed blood and worked hard for years to provide Iraqis the opportunity to have their own governments."
Kerry, who is expected to travel to Iraq soon at the request of President Barack Obama, will also discuss possible oil disruptions from the Iraq conflict with Gulf oil producers this week, a senior State Department official said.
"Egypt and the United States share the deep worries about the ongoing situation in Iraq and it is important that the two countries and their Gulf partners coordinate to face the challenges and risks," Shukri said.
CONCERN OVER ABUSES
Kerry is the highest-ranking U.S. official to visit Egypt since Sisi, the former military leader who toppled Islamist President Mohamed Mursi after mass protests last year, won a presidential election in May.
His trip came a day after an Egyptian court confirmed death sentences against 183 members of Mursi's Muslim Brotherhood, including its leader Mohamed Badie, in a mass trial on charges of violence in which one policeman was killed.
The Obama administration has said it looks forward to working with Sisi's government but expressed concern over widespread rights abuses and limits on freedom of expression.
"This is a critical moment of transition in Egypt (and) enormous challenges," Kerry said shortly after arriving. "There are issues of concern ... but we know how to work at these."
Shukri told his American counterpart that the relationship between the U.S. and Egypt should be based on "mutual respect and shared interests and with no interference in internal affairs," according to a statement by the Foreign Ministry.
A senior State Department official said Washington was worried about the use of heavy-handed political and security tactics by Egypt's authorities, which he said was polarizing Egyptian society.
"They in some ways are radicalizing certain aspects of Egyptian society in ways that are not supportive of overall stability in Egypt," said the official.
Still, there had been "a few flickering signs of positive movement" in recent weeks. Among these was the release of an Al Jazeera journalist, steps to start addressing sexual violence against women and Sisi's call during his first cabinet meeting for the revision of the human rights law.
The United States, which has counted on Egypt as a close Middle East ally for decades following its 1979 peace treaty with U.S. ally Israel, froze some of its $1.3 billion in annual military aid to Egypt following Mursi's overthrow.
About $575 million in suspended funds have been released over the past 10 days and will be used to pay existing defense contracts, the State Department official said. Washington has also said it will provide 10 Apache attack helicopters to help soldiers battling militants in the Sinai peninsula.
The Obama administration has made clear that the remaining funds, which require Congressional approval, will be released once there is evidence Sisi's government is taking further steps towards democracy, the senior State Department official said.
He described the mass trials and death sentences of Muslim Brotherhood supporters as a serious issue and suggested the courts were taking their lead from the country's rulers.
"The judiciary is responding to a political environment that the government has created," the official said, adding the United States did not believe that the Muslim Brotherhood posed a security threat to Egypt and had seen no information that substantiated a link to terrorist groups.
"We believe that in a general sense the Egyptian government needs to have a politically inclusive approach, which means that they need to include, and find ways to reach out to, the Muslim Brotherhood," the official said. (Additional reporting by Yasmine Saleh. Editing by Dominic Evans and Rosalind Russell)