The Muslim Brotherhood - the biggest group in parliament - called the protest after Suleiman announced his candidacy last week. His presidential bid has alarmed reformists, who regard him as a threat to their hopes for democratic change.
"Suleiman, do you think this is the old days?" chanted the protesters gathered in the square, the cradle of the uprising where Egyptians last year united to sweep Mubarak from power but which on Friday was mostly filled by Islamists alone. Others boycotted, reflecting deep divisions in the reform movement.
Muslim Brotherhood supporters waved the group's green flag and the red, white and black Egyptian national colors. "The people demand the fall of the regime!" they chanted, a slogan heard during the anti-Mubarak revolt. They also sang the national anthem and chanted "Down, down with military rule".
Several thousand Islamists protested in the northern coastal city of Alexandria. "I reject any replication of the old regime," said Taher Ismail, 42, one of the protesters.
Banners in Tahrir Square showed Suleiman and Mubarak alongside the Star of David, depicting both as agents of Israel - a perception stemming from policies that included Egypt's role in enforcing a blockade on the Hamas-run Gaza Strip.
Suleiman played a major role in managing a Middle East policy which became the focus of ever sharper public criticism during Mubarak's last years in power, long after Egypt under his predecessor Anwar Sadat made peace with Israel in 1979.
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 last year.
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.
Reformists fear his candidacy represents an army manoeuvre to keep control of the post held by ex-military men since the monarchy was overthrown in 1952 - an assertion denied by both the military and Suleiman.
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 protest groups that led the anti-Mubarak uprising. They oppose the Suleiman presidential bid but are also angry at the Islamists who they say have put the pursuit of power above the goals of the revolution, and they are planning their own protest next Friday.
One youth movement said in a statement they would take part in a protest next Friday. "It is time to abandon the Brotherhood as they abandoned the revolution and the youth," it said.
"The Brotherhood are going down for their own organizational reasons: for the goals of the Brotherhood and not the public," Ahmed Maher, one of the founders of the April 6 movement that helped ignite the revolt last year, told Reuters.
(Additional reporting by Ali Abdelati and Abdel Rahman Youssef in Alexandria; Writing by Tom Perry; Editing by Mark Heinrich)
Reuters