Tens of thousands have gathered in Cairo's Tahrir Square to protest against a decision by the ruling military council to assume new powers.
The protests were called by the Muslim Brotherhood, as it claims its candidate won last weekend's presidential poll.
His rival, former PM Ahmed Shafiq, also says he has won.
Meanwhile officials said ex-President Hosni Mubarak, ousted in last year's revolution, had been moved from jail to a military hospital after a stroke.
As Egyptians voted, the generals dissolved parliament and claimed all legislative power for themselves.
Correspondents say the Supreme Council of the Armed Forces (Scaf) appears to be working on the assumption that Muslim Brotherhood Mohammed Mursi will win, and making moves designed to reduce or constrain the power of the president and entrench its own.
Activists have described the moves as a "military coup".
Demonstrators have been chanting slogans against the military council, in the same square where huge protests last year led to the fall of President Mubarak.
Youth activists and liberals, many of whom refused to take part in the election run-off, are also involved in the protests.
But the BBC's Kevin Connolly in Cairo says that this year the message is more complex.
"[The election is] not totally stolen, but they have put some obstacles to fully transfer the power of the revolution and the voice of the street, and the voice of the critical mass to rule the country," one protester told the BBC.
But our correspondent says there are plenty of Egyptians who seem to be reconciled to the prospect of the military continuing to exercise power because they fear that otherwise the Muslim Brotherhood might turn their secular society into something resembling the theocracy of Iran.
As the protests were going on, Egyptian state TV reported that Mubarak had suffered a stroke, and security officials said he was moved from prison hospital to a military facility.
He was admitted to intensive care at the Maadi armed forces hospital, state media said.
The 84-year-old was sentenced to life in prison for his role in the deaths of protesters during last year's revolution, and there have been frequent reports since then about his deteriorating health.
The BBC's Jon Leyne in Cairo says Egyptians will be very sceptical about any reports about the former presidents health, which have often proved wrong in the past, and sometimes have an ulterior motive.
But these reports are better sourced than any before, he says.
Earlier the Muslim Brotherhood claimed victory in the election.
A spokesman for its Freedom and Justice Party (FJP) announced its chairman, Mohammed Mursi, had received 51.74%, compared with former Prime Minister Ahmed Shafiq, who he said had 48.26%.
The Higher Presidential Election Commission (HPEC) is scheduled to announce the official results on Thursday, but it usually shares them with the candidates beforehand to give them a chance to make objections.
The results tally with what Egyptian media and independent observers have been reporting, and the FJP's own calculations, which it released on Monday.
However, Mr Shafiq's campaign vigorously denied its candidate had lost.
In a news conference shown on Egyptian television, representatives of Mr Shafiq's campaign said the papers that Mr Mursi's campaign referred to did not come officially from the HPEC, and insisted that Mr Shafiq was ahead in the tally.
"We are willing to do whatever necessary at a legal level to prove he's the next president," Shafiq campaign spokesman Karim Salem said.
Troops outside parliament
Voting over the weekend to choose a successor to Hosni Mubarak, who was forced to step down by last year's uprising, was overshadowed by two Scaf decrees.
The first ordered the immediate dissolution of parliament following Thursday's Supreme Constitutional Court ruling that the law governing the recent elections for the lower house was unconstitutional because party members had been allowed to contest seats in the lower house reserved for independents.
Troops were deployed outside the parliament building before the decree was issued on Saturday to prevent MPs gaining access. The FJP and the ultraconservative Salafist Nour party dominate both chambers.
The second decree, which was published after the polls closed on Sunday, amended the March 2011 constitutional declaration and gave the generals complete control over legislation and military affairs until fresh parliamentary elections are held.
The Scaf will also play a significant role in running the 100-member assembly that will draft the country's new constitution.
The new president - who will take office without the oversight of a parliament and without a permanent constitution to define his powers or duties - will be able to form and dismiss a government, ratify and reject laws, and declare war, but only with Scaf's approval.