The demonstration, called for by several of the groups that participated in the February 2011 uprising, comes after the military reclaimed legislative power following a court ruling dissolving the Islamist-led parliament.
The move, which came in a constitutional declaration that also saw the ruling military council claim the ability to veto articles in any new constitution, has overshadowed the country's presidential poll.
But uncertainty also reigns over the winner of that vote, with Muslim Brotherhood candidate Mohammed Morsi claiming an early victory, while his rival, ex-prime minister Ahmed Shafiq has refused to concede.
A confirmed win for Morsi would mark the first time the Islamists have taken the presidency of the Arab world's most populous nation.
But Shafiq's campaign has accused the Brotherhood of trying to steal victory, calling their claim of a win an "act of piracy" and urging people to wait for the official results expected on Thursday.
Egypt's state media reported that initial counts showed Morsi ahead.
"After the counting was finished in all of Egypt's 27 provinces, indications show that Mohammed Morsi has won 51 percent and Ahmed Shafiq won 49 percent," the state-owned Al-Ahram newspaper said on its website.
There were scenes of jubilation at Morsi's Cairo headquarters, where the candidate thanked voters and pledged to work "hand-in-hand with all Egyptians for a better future, freedom, democracy, development and peace."
"We are not seeking vengeance or to settle accounts," he said, adding that he would build a "modern, democratic state" for all Egyptians, Muslims and Christians alike.
Shafiq campaign officials refused to concede victory, saying their figures showed their man was ahead.
"It's a stolen victory because you can't claim to have won a presidential election while the polling stations are still closing," Shafiq campaign manager Ahmed Sarhan told reporters.
"It's an act of piracy to claim victory using totally false figures," he said, adding that preliminary results obtained by the campaign showed Shafiq "still ahead in the vote, with between 51 and 52 percent."
Whoever is declared the winner, the country faces the prospect of a looming showdown between the Brotherhood and the ruling military.
"The Brotherhood and the military council prepare for the battle of the parliament," the headline of the liberal Wafd newspaper read Tuesday.
The Brotherhood insists the parliament still has the power to legislate and said Monday that it would take part in "all popular activities against the constitutional coup and the dissolution of parliament, beginning on Tuesday."
The ruling military body, the Supreme Council of Armed Forces, has introduced de facto martial law, given itself control of the legislature and state budget and also granted itself veto power on a new constitution.
Revolutionary youth movements, which had been split over whether to boycott the election or to vote Shafiq out, denounced the declaration as a "coup."
"The military council, with its unconstitutional coup, gave itself (unprecedented) powers. The military council has never and will never recognise popular legitimacy that contradicts it," the Coalition of Revolution Youth said in a statement.
The Brotherhood has vowed to challenge the constitutional declaration, even as it also faces a lawsuit challenging its legitimacy and legal status.
The military on Monday sought to quiet fears over the declaration, insisting it will transfer power to the new president on schedule by the end of this month.
The election victor will swear his oath before the constitutional court by "June 30, this month," Mamduh Shahin, one of the ruling generals, told a news conference Monday.
Another SCAF general, Mohammed al-Assar, said the vote winner will enjoy full presidential powers.
"The president of the republic will be vested with all the powers of the president of the republic," Assar told reporters.