* Supporters of deposed president call for more protests

* United States calls for Mursi's release

* More conflict expected in lawless Sinai region (Recasts with prosecutors announcing investigation)

By Yasmine Saleh and Peter Graff

CAIRO, July 13 (Reuters) - Egypt announced a criminal investigation on Saturday against deposed Islamist president Mohamed Mursi, with prosecutors saying they were examining complaints of spying, inciting violence and ruining the economy.

Egypt's first freely elected leader has been held at an undisclosed location since the army removed him from power on July 3, but has not yet been charged with any crime. In recent days Washington has called for him to be freed and for the authorities to stop arresting leaders of his Muslim Brotherhood.

The public prosecutor's office issued a statement saying it had received complaints against Mursi, eight other named Islamist figures including top Brotherhood leader Mohamed Badie, and others it did not identify.

The complaints are a first step in the criminal process, allowing prosecutors to begin an investigation that can lead to charges. Announcing the step was unusual: typically prosecutors wait until charges are filed before making public statements.

Badie and several other Brotherhood officials already face charges for inciting violence that were announced earlier this week, but most of them have not been arrested.

The prosecutors did not say who had made the complaints. Egyptian law allows them to investigate complaints from police or any member of the public.

Mursi's Brotherhood called on Saturday for more mass demonstrations after a huge march broke up peacefully before dawn, ending a week in which at least 90 people were killed.

VIGIL

The Brotherhood, which has maintained a vigil near a Cairo mosque since before the army removed Mursi on July 3, has said it will not leave the streets until he is restored to power.

The military says it deposed Mursi in a justified response to popular demand after millions of people demonstrated against him. The Brotherhood says it was a coup that reversed democracy.

Turmoil in the most populous Arab state has alarmed Western donors. Egypt sits astride the Suez Canal and has a strategically important peace treaty with Israel.

Large crowds of Brotherhood supporters finally dispersed before dawn on Saturday after marching through the streets into the early hours holding up pictures of Mursi at traffic lights.

Tens of thousands had turned out on Friday for what the Brotherhood called a "day of marching on".

Mursi's opponents say those demonstrations are still much smaller than the ones that brought him down. However, the Brotherhood has shown its organisational muscle by keeping its vigil running into a third week and bringing in coachloads of supporters from the provinces during the Ramadan fasting month.

At one stage overnight, demonstrators stood behind barbed wire shouting at soldiers a few dozen metres (yards) away.

"I am here to say 'no' to the military coup and 'yes' to Mursi, who I see as my legitimate president," said Ahmed Adel, a 22-year-old student, in downtown Cairo.

Senior Brotherhood figure Essam El-Erian, one of those who faces arrest, called on his Facebook page for more demonstrations on Monday.

"Egypt decides through the ballot box, through protests, mass marches and peaceful sit-ins," he said.

BLOODY WEEK

Friday's demonstration passed off peacefully, in contrast to deadly violence a week earlier when 35 people were killed in running battles between pro- and anti-Mursi demonstrators.

On Monday, 57 people were killed in clashes between the army and Mursi supporters near a Cairo barracks. The army says it was responding to an attack by terrorists; the Brotherhood says its partisans were massacred.

The United States refuses to say whether it considers the army takeover a "coup", which under U.S. law would require it to cut off aid including $1.3 billion a year in military support.

In recent days it has described Mursi's rule as undemocratic because of the vast popular protests against him, but also urged the authorities to release him and stop detaining his followers. Its wavering position has infuriated both sides.

Turmoil since a popular uprising toppled president Hosni Mubarak in 2011 has wrecked Egypt's economy, scaring away tourists and investors, draining hard currency reserves and making it difficult to import food and fuel, which the government distributes at heavily subsidised prices.

Rich Gulf Arab states Saudi Arabia, the United Arab Emirates and Kuwait, happy at the overthrow of the Brotherhood, have offered Egypt $12 billion in cash, loans and fuel.

State news agency MENA said a shipment of 70,000 tonnes of diesel arrived in Alexandria on Saturday from Turkey and Sweden.

Egypt's interim authorities have set out a "road map" to restore full civilian rule, setting out plans for a new constitution and parliamentary elections in about six months, to be followed by a presidential vote.

A judge has been named interim president and a liberal economist, Hazem el-Beblawi, has been appointed prime minister.

He is trying to cobble together a cabinet likely to be made up mainly of technocrats and liberals, without offending a large ultra-orthodox Islamist group that broke with the Brotherhood to accept the military takeover. Beblawi told Reuters on Saturday he expected to name the cabinet within two days.

Egypt's crisis has raised fears over security in the lawless Sinai peninsula bordering Israel and the Palestinian Gaza Strip, where militants attack security forces checkpoints almost daily.

The headline in the state-run Al Gomhuria newspaper read: "Sinai Purification Operation within Days", referring to a possible army offensive against militants in the region.

But a senior army officer, who asked not to be named, said this was unlikely to take place immediately because forces are now focused on keeping the peace during political turmoil.

Security sources said on Saturday a Palestinian had been arrested in Egypt on suspicion of carrying out attacks on a natural gas pipeline. (Additional reporting by Shadia Nasralla and Omar Fahmy in Cairo and Yusri Mohamed in Ismailia; Writing by Peter Graff; Editing by Andrew Roche)