Egypt's leading Islamic institution has warned of a possible civil war as clashes between supporters and opponents of President Mohamed Morsi spread across the country on Saturday – exactly a year after his inauguration as the first democratically elected president.

Egypt's fate feels as uncertain as at any point since the 2011 uprising, which toppled Hosni Mubarak, with repeated rumours of military intervention.

At least eight people have died and more than 600 have been injured in fighting between Morsi's Islamist allies – who argue that his democratic legitimacy should be respected – and his often secular opponents, who say that he has not shown respect for the wider values on which a successful democracy depends.

But with at least four attacks nationwide on offices of the Muslim Brotherhood – the Islamist group from which Morsi hails – divisions are being increasingly drawn on ideological grounds. Anger is directed not just at Morsi, but at the Brotherhood, which is considered a partner in his eventual aim of restructuring the state along more religious lines. A former Brotherhood MP was killed in fighting earlier in the week.

The US has warned its citizens to defer non-essential travel to Egypt and told non-emergency diplomatic staff to leave. Cairo's main airport was crammed on Saturday, with all international flights reportedly fully booked.

"We're here to bring down the Murshid's regime," said Saad el-Aswany, a builder from Aswan attending an anti-Morsi protest in Tahrir Square. The Murshid is the Muslim Brotherhood's leader, and members of the opposition think his office is the real power behind Morsi's throne. "Banish the Murshid and all who are with him," shouted thousands alongside him, many waving Egyptian flags.

Five miles away, hundreds of thousands were waving the same national flag – but with a different vision of what it represented. "Islam, Islam," chanted those who had descended on the capital to support their president. "Islam in spite of liberalism."

"I'm here for Islamic law first, and democracy second," said Moustapha Sabry, a maths teacher who follows the strict Salafi form of Islam.

Cairo remained relatively calm on Saturday as tens of thousands of people, both supporters and opponents of Morsi, were engaged in relatively peaceful sit-ins. But with deaths in several provincial cities, there are mounting fears over stability — especially since Morsi's opponents have promised to mobilise millions on Sundayto force him from office. Protest organisers said a petition calling on Morsi to quit had collected 22 million signatures, although the figure, which amounts to 40% of the electorate, could not be verified.

President Barack Obama urged both sides to engage in dialogue and refrain from violence. If unchecked, he warned, the unrest could spill beyond the country's borders. "Egypt is the largest country in the Arab world," Obama said. "The entire region is concerned that, if Egypt continues with this constant instability, that has adverse effects more broadly."

Egypt's highest Islamic authority expressed its own concerns over the situation. "Vigilance is required to ensure we do not slide into civil war," said Sheikh Hassan al-Shafie, a senior cleric at al-Azhar, a 1,000-year-old mosque and university often considered the highest seat of learning in the Sunni Islam world.

"Egypt's current crisis goes beyond a reasonable political struggle that could be explained within the context of democratic dispute," argued Khalil al-Anani, an academic specialising in Egyptian politics. "It is more an attempt to banish and abort one party by the other."

But there were signs that the divisions were not unbridgeable. "I'm not here for religious reasons, I'm just here to respect the office of the presidency," said Tariq Shabasy, a lawyer and former member of Kefaya, the liberal Mubarak-era protest movement – a surprising attendee at the pro-Morsi rally.

Neither can Egypt's differences be solely explained along zero-sum lines. "This is not a binary struggle between Islamists and secularists," wrote Michele Dunne, director of the Atlantic Council's Rafik Hariri Centre for the Middle East. "Rather, there are at least three principal political forces at work: Islamists, secularists, and the old state."

Morsi's opponents include both the liberals and leftists who helped to bring down Mubarak and Mubarak-era officials who spent years repressing the Brotherhood and its allies. On Saturday, the Brotherhood sought to blame members of Mubarak's former party, the National Democratic party, for stoking opposition to Morsi. Opposition activists rejected this, though they condemned any hypothetical act of trouble-making by old-regime figures.

Morsi's opponents also disagree on who should follow him, were he to go. A senior military source indicated that it might step in should anti-Morsi protests rival those that toppled Mubarak. This would please the hundreds of Morsi opponents camped outside the ministry of defence calling for army intervention – but would appal many of the revolutionaries who rose to prominence during the 2011 uprising.

In a sign of the opposition's divisions, Tarek Shalaby, a hero of the 2011 uprising, was escorted from Tahrir square for querying military intervention.