The cross was erected after Kaczynski was killed in a plane crash in Russia in April, and has become a site of mourning for a small group of elderly supporters of the late leader and his brother, Jaroslaw Kaczynski, the head of a conservative pro-Catholic party who made an unsuccessful run for the presidency this summer.
It has also become a source of friction with their political rivals.
Those at the rally late Monday said they want the cross moved to a nearby church, arguing that it has no place in front of the presidential palace of a nation constitutionally defined as secular.
While church and state are technically separated in staunchly Catholic Poland, the church wields some influence in political life.
Authorities tried to move it last week but were prevented from doing so by praying demonstrators, raising the stakes on a cultural battle that pits a deeply conservative, often older constituency against an increasingly secular and younger population.
The protesters on Monday, many of them young people who were mobilized on Facebook, said they object to what they see as a small, ultra-religious minority defying the country's authorities and laws.
"If this happened in Germany or France, the police would just remove the cross. But here you have blackmail," said Michal Buczynski, a 28-year-old economist.
One banner said "Poland is a country of the rule of law" — a reference to the their view that the defenders of the cross are defying the country's laws and that authorities are powerless to implement them.
A small group of supporters of the cross held rosaries and prayed. They were separated by police from the thousands of protesters, some of whom mocked them, laughing and chanting that the cross should be moved "to the church."
A 32-year-old woman in the group said she came out just to observe the protest and that she was embarrassed by both sides.
"I am a Catholic and for me the cross is important. The people defending it have no right to do this because it's a public space, and they are hypocrites," said the woman, who would only give her first name, Dorota, because she works for a government office. "I don't understand why our authorities didn't move the cross a long time ago. But the behavior of the young people is bad too because they are saying bad things about the cross."
Scouting groups erected the simple wooden cross in the days after the April 10 plane crash. After authorities said they planned to move the cross to the nearby church of St. Anne's in Warsaw's historic center, the defenders of the cross have kept a nightly vigil aimed at preventing its removal. Recently even priests supporting the removal have been rebuffed by the cross's defenders.
Jaroslaw Kaczynski placed a wreath at the cross on Tuesday morning, exactly four months after the plane crash. Those holding vigil chanted "Jaroslaw! Jaroslaw!"
Kaczynski was defeated in an presidential race in July by Bronislaw Komorowski, a moderate and pro-European member of the governing Civic Platform party. Komorowski wants the cross gone. He has not moved into the palace yet because it is undergoing renovation.
SIC: AP