Police fired one round of tear gas before taking cover inside the palace grounds, The New York Times reported.
The march came after an Egyptian judicial council agreed to oversee a vote on the document. The decision by the Supreme Judicial Council -- which last month condemned a power grab by President Mohamed Morsi as an "unprecedented attack" on the courts -- suggests the influential judicial organization is looking to contain Egypt's political unrest, the Los Angeles Times said.
The Muslim Brotherhood, with which Morsi is affiliated, quickly endorsed the council's move.
Many judges are on strike and vowed to boycott the referendum, scheduled for Dec. 15.
The demonstration by Morsi's political opponents was intended to pressure him to rescind a Nov. 22 decree that gave him immunity from judicial oversight. Morsi has said he needs near-absolute power to speed up Egypt's democratic transition. He has said the decree will be nullified when the charter is adopted.
The biggest march Tuesday -- led by leading dissidents, including Nobel laureate Mohamed ElBaradei -- was dubbed "The Final Warning." It was also intended to protest the charter itself, which the April 6 youth movement said "threatens the freedoms and rights of Egyptians."
The draft Constitution provides for freedom of expression but prohibits "insults" to "religious prophets" and establishes that news media must uphold public morality and the "true nature of the Egyptian family." It requires television stations and websites to receive authorization from the government before they can operate.
Secular and liberal opponents of the draft Constitution say it will allow Islamists to push Egypt toward religious conservatism, The New York Times said.
Heba Morayef, a Human Rights Watch researcher, said the document "fatally" undermines the protection of freedom of expression by limiting expression.
"It is designed to let the government limit those rights on the basis of 'morality' or the vague concept of 'insult,'" Morayef said.
Eleven newspapers did not publish Tuesday and at least three television networks said they plan to go dark Wednesday in protest of what they call restrictions of freedom of expression contained in the draft constitution, the newspaper said.