(Adds French reaction, UN call for investigation)
* Protests after Friday prayers could trigger new violence
* Muslim Brotherhood says not ready to retreat in battle
* Obama says deplores violence, cancels Egypt war games
By Crispian Balmer and Yasmine Saleh
CAIRO, Aug 16 (Reuters) - Deeply polarised Egypt braced for renewed confrontation on Friday after the Muslim Brotherhood called for a nationwide march of millions to show anger at a ferocious security crackdown on Islamists in which hundreds were killed.
Defying criticism from major Western allies, Egypt's army-backed government warned it would turn its guns on anyone who attacked the police or public institutions after protesters torched a government building in Cairo on Thursday.
At least 623 people died and thousands were wounded on Wednesday when police cleared out two protest camps in Cairo set up to denounce the military overthrow on July 3 of Egypt's first freely elected president, Islamist leader Mohamed Mursi.
It was the third mass killing of Mursi's supporters since his ouster. The assault left his Muslim Brotherhood in disarray, but it said it would not retreat in its showdown with army commander General Abdel Fattah al-Sisi.
"After the blows and arrests and killings that we are facing, emotions are too high to be guided by anyone," said Brotherhood spokesman Gehad El-Haddad.
A Brotherhood statement called for a nationwide "march of anger" by millions of supporters on Friday after noon prayers.
"Despite the pain and sorrow over the loss of our martyrs, the latest coup makers' crime has increased our determination to end them," it said.
The Brotherhood accuses the military of staging a coup when it ousted Mursi. Liberal and youth activists who backed the military saw the move as a positive response to public demands.
Friday prayers have proved a fertile time for protests during more than two years of unrest across the Arab world.
In calling for a "Friday of anger," the Brotherhood used the same name as that given to the most violent day of the 2011 uprising against former President Hosni Mubarak. That day, Jan. 28, 2011, marked the protesters' victory over the police, who were forced to retreat while the army was asked to step in.
In a counter move, a loose liberal and leftist coalition, the National Salvation Front, called on Egyptians to protest on Friday against the Brotherhood's "obvious terrorism actions".
Signalling his displeasure at the worst bloodshed in Egypt for generations, U.S. President Barack Obama said on Thursday normal cooperation with Cairo could not continue and announced the cancellation of military exercises with Egypt next month.
"We deplore violence against civilians. We support universal rights essential to human dignity, including the right to peaceful protest," he said, but stopped short of cutting off the $1.55 billion a year of mostly military U.S. aid to Egypt.
The United States on Thursday told its citizens to leave Egypt due to the unrest. It issued the same advice last month.
The Egyptian presidency issued a statement saying Obama's remarks were not based on "facts" and would strengthen and encourage violent groups that were committing "terrorist acts".
Despite its aid to Egypt, Washington's influence over Cairo has been called into question during the turmoil, which has seen Saudi Arabia, Kuwait and the United Arab Emirates pledge $12 billion in assistance, making them more prominent partners.
By cancelling the military exercise, but not cutting off U.S. aid, Obama was seeking to show his displeasure at the violent crackdown without totally alienating the generals.
U.S. Defense Secretary Chuck Hagel issued a statement saying he had called Sisi on Thursday to say Washington would maintain its military relationship with Egypt, but he also told him the recent violence was putting defence cooperation at risk.
"Since the recent crisis began, the United States has made it clear that the Egyptian government must refrain from violence, respect freedom of assembly, and move toward an inclusive political transition," Hagel said.
"I reiterated that the United States remains ready to work with all parties to help achieve a peaceful, inclusive way forward."
The White House has voiced support for democracy in Egypt, while seeking to protect the U.S. strategic interest in Egypt's stability, its peace treaty with Israel and cooperation with the U.S. military, including privileged access to the Suez Canal.
Critics argue that Obama had done too little, too late and that his administration has repeatedly sent mixed messages - among them its failure to brand Mursi's ouster a military coup - thereby eroding its ability to influence events.
France's foreign minister said on Friday the escalating tension in Egypt risked playing into the hands of radical groups and urged all sides to do their utmost to defuse the tension.
"Given that Egypt was the guarantor of peace in the region, it's even more worrying," Laurent Fabius told RTL radio.
"Maximum restraint must be shown otherwise the risk is that extremist groups take advantage of the situation and that would be extremely serious."
The Egyptian press was full of praise for security forces, illustrating the Gulf between Cairo and its Western allies.
By comparison with Western criticism, the UAE said Egypt's government had "exercised maximum self-control".
The Arab nations' cash, which started arriving in July, is aimed at stabilising Egypt's wobbling economy, which is suffering from a ballooning budget deficit and high inflation.
This week's carnage will further damage state coffers. The government has imposed a nighttime curfew set to last at least a month, a move that will hit the crucial tourism industry.
On Thursday, the U.N. Security Council urged all parties in Egypt to exercise maximum restraint.
"The view of council members is that it is important to end violence in Egypt," Argentine U.N. Ambassador Maria Cristina Perceval said after the 15-member council met on the situation.
Turkish Prime Minister Tayyip Erdogan had earlier also called for the U.N. Security Council to convene quickly after what he described as a massacre in Egypt and criticised Western nations for failing to stop the bloodshed.
U.N. human rights chief Navi Pillay has called for an independent investigation into Wednesday's events in Egypt.
"Egypt is party to the International Covenant on Civil and Political Rights, and as such has international legal obligations to investigate human rights violations. So that's what underpins our call," U.N. spokeswoman Elizabeth Throssell told a news briefing in Geneva on Friday.
On the unusually quiet streets of Cairo, there was little sympathy for the Brotherhood, which won all five elections after Mubarak's fall in 2011 but which was accused of incompetence and partisanship during Mursi's brief time in charge.
"We didn't want this to happen, but at the end of the day they pushed us to do it," said Mahmoud Albaz, 33, an actor and real-estate agent who lives near the Brotherhood protest camp at the Rabaa al-Adawiya mosque, now blackened by fire and soot.
Many of those who died on Wednesday were still laid out more than a day later in Cairo morgues and at a city mosque. Their families accused the government of putting bureaucratic hurdles in their way to make it hard to obtain permission to bury them.
Under Islamic tradition, bodies ought to be buried within 24 hours of death.
"We arrived at 7 a.m. The whole family is here," said Atif Hashim, a 50-year-old teacher, as he waited on Thursday to collect the body of his cousin, a father of five young children.
"They just drink tea inside, they just throw the bodies on the floor with some ice," he said of officials in the morgue. (Additional reporting by Shadia Nasralla, Michael Georgy, Tom Finn, Tom Perry and Ahmed Tolba in Cairo, Steve Holland and Jeff Mason in Martha's Vineyard, Massachusetts, Andrea Shalal-esa in Washington and Stephanie Nebehay in Geneva; Editing by Alistair Lyon)