The Muslim Brotherhood - the biggest group in parliament - called the protest after Suleiman announced his candidacy last week. Both Islamists and secular reformists view it as a threat to democratic reform.
"Suleiman, do you think this is the old days?" chanted the protesters gathered in the square, the cradle of the uprising that swept Mubarak from power last year.
Muslim Brotherhood supporters waved the group's green flag and the red, white and black Egyptian national colours. "The people demand the fall of the regime," they chanted, a slogan used during the anti-Mubarak uprising. "Down, down with military rule," they chanted. They also sang the national anthem.
Banners showed Suleiman and Mubarak alongside the Star of David, depicting both as agents of Israel because of policies that included Egypt's role in enforcing a blockade on the Hamas-run Gaza Strip, which borders the Arab state.
Egypt has had a peace treaty with Israel since 1979 but Mubarak's Middle East policy, in large part managed by Suleiman, became the focus of ever sharper public criticism in his last years in power.
The council of army generals that has been running Egypt since Mubarak was deposed is due to hand power to an elected president on July 1. The vote, Egypt's first real presidential election, is due to get under way on May 23 and will likely go to a run-off in June between the top two candidates.
Frontrunners include the Muslim Brotherhood's Khairat al-Shater, ultra-orthodox Salafi sheikh Hazem Salah Abu Ismail, ex-Arab League Secretary General Amr Moussa and Ahmed Shafiq, who served as prime minister in Mubarak's last days in power.
The Islamist-dominated parliament on Thursday passed legislation that would stop both Suleiman and Shafiq from running on the grounds they served in top posts under Mubarak. However, analysts doubt the law will be enacted by the ruling generals, setting the stage for more tension.
In an interview with the state-run al-Ahram newspaper, Suleiman pledged to press ahead with his campaign.
'Complete the march'
"I am confident and have all faith that we will complete the march and this type of law will wreck the country, especially with the dominance of the Brotherhood over everything," he said.
Mubarak appointed Suleiman as his vice president in his last days in power. Suleiman, 74, publicly engaged the Brotherhood and other opposition forces during a failed effort to quell the uprising.
An army general, he is closely associated with the security policy of a state that kept the Islamists on a tight leash, maintaining an official ban on the Brotherhood and deploying heavy force against more radical Islamists who took up arms.
Suleiman and the military deny claims that his candidacy represents an army bid to keep control of the post held by ex-military men since the monarchy was overthrown in 1952.
Shater, the Brotherhood candidate, has described Suleiman's candidacy as an insult to Egyptians who rose up against Mubarak.
However Suleiman does appear to have a constituency among Egyptians alarmed by the rise of Islamists and who see him as the kind of strong man needed to restore stability after a year of political turmoil that has hit the economy hard.
There was no sign in the square of the reformist youth protest groups that led the anti-Mubarak uprising. They are angry at the Suleiman presidential bid but are also opposed to what they see as Islamist attempts at domination.
One youth movement said in a statement they would take part in a protest next Friday and boycott the Brotherhood-led protest. "It is time to abandon the Brotherhood as they abandoned the revolution and the youth," it said.