* Muslim Brotherhood calls march after week of bloodshed

* Washington continues to walk diplomatic tightrope

* Concerns prompt U.S. Navy to move ships nearer (Adds interim prime minister comments)

By Yasmine Saleh and Mike Collett-White

CAIRO, July 12 (Reuters) - Supporters of ousted President Mohamed Mursi called for protests on Friday and Egyptians prayed there would be no repeat of clashes that have killed more than 90 people in the last week in the bitterly divided Arab nation.

More than a week after the army toppled Egypt's first elected leader on a wave of demonstrations, Mursi's Muslim Brotherhood movement wants people to join it on the streets to push for his reinstatement, which now looks like a lost cause.

The streets of Cairo were quiet on Friday morning, with separate demonstrations by Mursi supporters and opponents expected later in the day, the weekly Muslim prayer day.

Officials say Mursi is still being held at the Republican Guard compound in Cairo, where troops killed 53 Islamist protesters on Monday in violence that intensified anger his allies already felt at the military's decision to oust him.

Four members of the security forces were also killed in that confrontation, which the military blames on "terrorists". Mursi's supporters call it a massacre and say those who died were praying peacefully when troops opened fire.

Many of Egypt's 84 million people have been shocked by the shootings, graphic images of which have appeared on state and private news channels and social media. The incident occurred just three days after 35 people were killed in clashes between pro- and anti-Mursi demonstrators across the country.

"It's a very hard time for Egyptians, to see footage of blood and violence during the holy month of Ramadan, and everyone I speak to says the same thing," said Fateh Ali, a 54-year-old civil servant in Cairo.

"I really hope the situation gets resolved soon. I don't think we can afford this economically or psychologically."

The Brotherhood contends it is the victim of a military crackdown, evoking memories of its suppression under Hosni Mubarak, whose 30-year rule collapsed 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.

The unrest has also raised fear over security in the lawless Sinai peninsula bordering Israel and the Palestinian Gaza Strip. One Egyptian policeman was killed and another wounded early on Friday when militants fired rocket-propelled grenades at checkpoints in the Sinai town of El Arish.

Egyptian state media said police arrested three Palestinian militants for attempted attacks in Sinai.

VIGIL, SONGS FOR THE DEAD

Outside the Rabaa Adawiya mosque in northeastern Cairo, thousands of Brotherhood supporters gathered late on Thursday to mourn the dead in Monday's violence, the deadliest since Mubarak was toppled, apart from a 2012 soccer stadium riot.

Women wailed and men cried as they watched a large screen showing graphic footage of hospital scenes immediately after the shooting, with corpses on the floor and medics struggling to cope with the number of bloodied casualties being carried in.

Hundreds of Egyptian flags fluttered in the evening breeze. Songs of defiance were sung. Many thousands of Islamists have camped out in the area, braving searing heat and, since Wednesday, daytime fasting during Ramadan.

It has become the de facto base of the Brotherhood, whose leaders live under the threat of detention after the public prosecutor ordered their arrests earlier in the week.

Judicial sources say authorities are expected to charge Mursi, possibly for corruption or links to violence. Prosecutors are also taking a fresh look at an old case over a 2011 prison break when Mursi was among Brotherhood figures who escaped after being rounded up during anti-Mubarak protests.

The detentions and threats of arrest have drawn concern from the United States, which has walked a semantic tightrope to avoid calling Mursi's ouster a military coup.

U.S. law bars aid to countries where a democratic government is removed in a coup. Washington, which gives Egypt's military $1.3 billion in aid each year, has said it is too early to say whether Mursi's removal by the army meets that description.

The army said it was enforcing the nation's will after millions of people, fed up at economic stagnation and suspicious of a Brotherhood power grab, took to the streets at the end of June to demand his resignation.

State Department spokeswoman Jen Psaki said on Wednesday Mursi's government "wasn't a democratic rule".

Her words were warmly received by the interim government and swiftly denounced by the Brotherhood. On Thursday, Psaki expressed concern over the crackdown on Brotherhood leaders.

"If politicised arrests and detentions continue, it is hard to see how Egypt will move beyond this crisis," she said.

ALARM OVERSEAS

Crucial to longer-term stability will be holding parliamentary and presidential elections, which the transitional authorities are hoping to achieve in a matter of months.

Adli Mansour, the interim president named by the general who removed Mursi, has announced a temporary constitution, plans to amend it to satisfy parties' demands and a faster-than-expected schedule for parliamentary elections in about six months.

He has named liberal economist Hazem El-Beblawi as interim prime minister. Beblawi told Reuters he would start contacting candidates for ministerial posts on Sunday and Monday, with a view to swearing in a cabinet next week.

Negotiations are difficult, with the authorities trying to attract support from groups that range from secularists to ultra-orthodox Muslims, nearly all of whom expressed deep dissatisfaction with elements of the interim constitution.

Underlining the level of concern overseas at Egypt's crisis, two U.S. Navy ships patrolling in the Middle East moved closer to Egypt's Red Sea coast in recent days, in what appeared to be a precautionary move following Mursi's ouster on July 3.

The United States often sends Navy vessels close to countries in turmoil in case it needs to protect or evacuate U.S. citizens or give humanitarian assistance.

Rich Gulf states have thrown Egypt a $12 billion lifeline in financial aid, which should help it stave off economic collapse.

More than two years of turmoil have scared away tourists and investors, shrivelled hard currency reserves and threatened Cairo's ability to import food and fuel. (Additional reporting by Alexander Dziadosz, Maggie Fick, Sarah McFarlane, Mike Collett-White, Tom Perry, Peter Graff, Ali Saed, Seham el-Oraby, Shadia Nasralla and Tom Finn in Cairo and Andrea Shalal-Esa and Lesley Wroughton in Washington; Writing by Mike Collett-White and Peter Graff; Editing by Alistair Lyon)