* Ashton meets general who overthrew Islamist president
* Visit follows Saturday killing of 80 Mursi supporters
* Mursi supporters march towards state security building
* Arrests, killings portend crackdown on Islamists (Adds march)
By Maggie Fick and Asma Alsharif
CAIRO, July 29 (Reuters) - Europe's top diplomat shuttled between Egypt's rulers and the Muslim Brotherhood on Monday in a mission to pull the country back from more bloodshed, but both sides were unyielding after 80 Islamist supporters were gunned down.
European Union foreign policy chief Catherine Ashton, making her second visit in 12 days as one of the few outsiders able to speak to both sides, made no public comment. Supporters and opponents of ousted Islamist president Mohamed Mursi left no doubt about the depth of polarisation in the Arab world's most populous nation.
"It's very simple, we are not going anywhere," said Brotherhood spokesman Gehad El-Haddad, making clear the movement intends to defy government orders to abandon a protest vigil by thousands of followers demanding Mursi's return.
"We are going to increase the protest," he told Reuters. "Someone has to put sense into this leadership."
Raising the prospect of more bloodshed erupting during Ashton's visit, the Brotherhood said it would march again on Monday evening from its month-old vigil at a mosque in northern Cairo towards offices of the Interior Ministry.
Several hundred Mursi supporters made good on the promise, marching towards a state security building in Cairo late at night. They chanted "down with military rule" and obstructed traffic on a main road near their protest camp.
Backers of the military that deposed Mursi on July 3 were equally unbending, despite Saturday's dawn carnage when security forces shot dead at least 80 Brotherhood supporters after a day of rival mass rallies.
"We asked her (Ashton), would you accept an armed sit-in under your roof?" said Mahmoud Badr, a leader of the Tamarud youth movement that mobilised huge protests against Mursi before the army moved against him.
"What if al Qaeda had a sit-in in a European country? Would you leave it be?" he asked reporters after meeting Ashton, echoing the army's branding of its opponents as terrorists.
U.S. MILITARY AID
The violence has raised global anxiety that the army may move to crush the Brotherhood, a movement which emerged from decades in the shadows to win power in elections after Egypt's 2011 Arab Spring uprising against Hosni Mubarak.
The White House, treading a fine line with a pivotal Arab ally and recipient of $1.3 billion a year in U.S. military aid, said on Monday it "strongly condemns" Saturday's bloodshed, and urged respect for the right to peaceful protest.
"Violence not only further sets back the process of reconciliation and democratisation in Egypt, but it will negatively impact regional stability," spokesman Josh Earnest said.
Ashton met General Abdel Fattah al-Sisi, the head of the army and the man behind the overthrow of Egypt's first freely-elected president. She also held talks with members of the interim government installed by the army, and representatives of the Freedom and Justice Party, the Brotherhood's political wing.
Thousands of Brotherhood supporters have been camped out for a month at Cairo's Rabaa al-Adawiya mosque, demanding Mursi's reinstatement and defying threats by the army-backed authorities to remove them.
Ashton was expected to speak to reporters on Tuesday. Before arriving, she said she would press for a "fully inclusive transition process, taking in all political groups, including the Muslim Brotherhood".
Her leverage is limited. The United States is Egypt's chief Western backer and source of military hardware, though the EU is the biggest civilian aid donor to the country, a strategic bridge between the Middle East and North Africa.
The EU has attempted to mediate in the political crisis over the past six months as Egyptians have grown increasingly suspicious of U.S. involvement. President Barack Obama delayed delivery last week of four F-16 fighter jets to Egypt, in a gesture of displeasure at the turn of events.
Mursi has been in detention since he was ousted and the military-backed interim government has placed him under investigation on charges that include murder.
The handling of his case by the military suggests it believes it has the support of a big majority of Egyptians.
Army chief Sisi has emerged as the public face of the new order, enjoying fawning coverage in Egyptian media and sowing doubts about the military's promise to hand over to full civilian rule with a "road map" to parliamentary elections in about six months.
A banner stretched across a road in downtown Cairo declared: "The army, the police, and the people - one hand".
Security forces shot dead dozens of Mursi supporters at dawn on Saturday when they marched from their vigil. The Health Ministry put the death toll on Monday at 80, up from 72. The Interior Ministry said one police officer had succumbed to his wounds on Monday. Nearly 300 people have died in violence since Sisi deposed Mursi.
Interim Interior Minister Mohamed Ibrahim has denied police culpability, saying his officers intervened with tear gas in clashes between the Brotherhood and angry local residents.
A group of Egyptian human rights groups called on Monday for his dismissal. The latest "massacre", they said, "joins a long list of killings documented by rights groups" since Mubarak's fall.
Saturday's bloodshed was the worst since July 8, when security forces killed more than 50 Brotherhood supporters outside a Cairo barracks. The army said its forces had fired back after being attacked; the Brotherhood said its supporters were praying.
The interim cabinet has vowed to clear the Brotherhood's mosque vigil after complaints from residents about the huge encampment on their doorstep. The Islamists vowed to keep marching.
"The danger we face because of the political situation and the coup is greater than the violence we face in marches," said Brotherhood member Islam Tawfiq, 26. (Additional reporting by Tom Perry, Michael Georgy, Matt Robinson, Yasmine Saleh and Shaimaa Fayed; Writing by Matt Robinson; editing by David Stamp)