By Yasmine Saleh and Mike Collett-White

CAIRO, July 13 (Reuters) - Tens of thousands of Egyptians marched on Cairo's streets in the early hours of Saturday to demand ousted Islamist President Mohamed Mursi be reinstated, but there were none of the deadly clashes that swept Egypt a week ago.

Mursi's Muslim Brotherhood called on Friday for "a day of marching on", and 10 days after the military overthrew Egypt's first freely elected president, large crowds descended on the capital waving flags and chanting slogans.

A week earlier similar scenes of protest turned violent when pro- and anti-Mursi demonstrators clashed in cities and towns across the country, killing 35 people and widening the rift between Egyptians on different sides of the political divide.

Three days after that, Egypt was left reeling again when soldiers opened fire on a group of pro-Mursi demonstrators outside the Republican Guard compound in Cairo where he is being held, killing 53. Four soldiers also died.

The powerful military blamed the confrontation on "terrorists". Mursi's supporters call it a massacre and say those who died were praying peacefully when troops opened fire.

As midnight passed in Cairo, large crowds of Brotherhood supporters were still out. Near the Ministry of Defence, hundreds of demonstrators standing behind barbed wire shouted at soldiers standing a few dozen metres away.

"I am here to say 'no' to the military coup and 'yes' to Mursi, who I see as my legitimate president, although I am not in the Brotherhood and nor did I vote for him," said Ahmed Adel, a 22-year-old student, in downtown Cairo.

The army denies Mursi's overthrow was a coup, saying it ousted him to enforce the will of the people after millions took to the streets at the end of June to demand his resignation.

The Brotherhood contends it is the victim of a military crackdown, evoking memories of its suppression under autocrat Hosni Mubarak, toppled in an uprising in 2011.

But many of its opponents blame Islamists for the violence, and some have little sympathy for the demonstrators who died, underlining how deep the fissures in Egyptian society are.

DEFIANCE AT VIGIL

At a Cairo mosque where Mursi supporters have held a mass vigil for more than two weeks, crowds swelled as people were bused in from the provinces, where the Brotherhood has strongholds.

Amer Ali, a member of parliament who spent 13 years as a political prisoner under Mubarak, made the five-hour car journey from Assiut in the Nile valley with his wife and two children.

"We're here and we're not leaving," he said. "We came with our kids to support legitimacy, democracy, and ... the first freely elected president in the Arab world."

People squirted water from bottles to cool each other down. Dozens rested in the shade of tents, dozing or reading the Koran, conserving energy during the Ramadan fasting month when Muslims refrain from food and water during daylight.

Passions sometimes ran high.

"They killed our martyrs while they were praying!" screamed Soraya Naguib Ahmed, tears down her face visible through the slit of her full-face veil, referring to the clash on Monday.

Mursi remains in detention at the compound outside which the shooting took place, and judicial sources said he was likely to be charged soon, possibly for escaping prison during the 2011 uprising when authorities arrested many Brotherhood leaders. Mursi could also be charged for corruption or links to violence.

Asked whether Washington agreed with the German Foreign Ministry's call for Mursi to be released, U.S. State Department spokeswoman Jen Psaki said: "We do agree."

She declined to say if the United States had officially conveyed its wish to Egyptian officials and the military.

Egypt's foreign ministry said in a statement it was committed to a political "road map" drafted by the army which sets out a timetable for fresh parliamentary and presidential elections, possibly within months.

In reaction to international calls for Mursi's release, it added that Egypt's interim authorities would not revert to any "exceptional or illegal measure."

SINAI CONCERNS

The unrest has raised fears over security in the lawless Sinai peninsula bordering Israel and the Palestinian Gaza Strip.

Militant groups in North Sinai have promised more attacks and urged Islamists to take up arms in the region, which is near the Suez Canal, a vital waterway linking Asia and Europe.

The state-controlled Al Gomhuria newspaper ran a front page headline: "Sinai's Purification Operation in Days", referring to expectations the army would launch an offensive against militants in the region.

There have been almost daily attacks and skirmishes between radical Islamists and police and soldiers in Sinai since Mursi's ouster, some of them deadly.

Overnight on Friday, gunmen fired on a security checkpoint near the Suez Canal, but security personnel repelled the attack.

Mohab Mamish, Chairman of the Suez Canal Authority, told Reuters the shooting had no impact on the flow of ships.

"The Canal is very well secured and the attack was away from it and any of its administrative buildings," Mamish said.

The Suez is vital to Egypt's economy, decimated by two and a half years of turmoil since Mubarak was toppled.

Foreign reserves and wheat stocks are running low, although financial aid of $12 billion from three wealthy Gulf states is likely to ease the crisis in the short term.

In the longer term, the transition from the military-backed interim authorities to an elected parliament and president will be crucial to Egypt's stability.

Interim Prime Minister Hazem el-Beblawi said on Friday he had named centre-left politician and lawyer Ziad Bahaa el-Din as his deputy, and he expected to swear in a cabinet next week. (Additional reporting by Ali Abdelatty, Omar Fahmy, Peter Graff, Tom Finn, Maggie Fick, Noah Browning and Ulf Laessing in Cairo and Yusri Mohamed in Ismailia; Writing by Mike Collett-White; Editing by Eric Walsh)