The current constitution enshrines President Bashar Assad's Baath as the leader of the state and society.
Under the draft passed Sunday night, which would need approval by Parliament, any party can be set up if it is not based on religious, tribal or ethnic factions or discriminates by gender or race, the state-run SANA news agency reported.
The Syria Report economic newsletter said the law seemed crafted to bar Kurdish and Islamic parties.
Louay Hussein, an opposition figure in Damascus, told The New York Times: "A new law is not going to stop the government from violating our personal and political freedoms. So that law does not really have any significance."
Clashes broke out Sunday in Beirut, Lebanon, between supporters and opponents of Assad, officials said.
Lebanese forces broke up the fighting, which followed a peaceful demonstration by supporters of the Assad government, The Daily Star reported.
About 2,000 demonstrators rallied at the Syrian Embassy in Beirut to show support for Assad, who has been facing pro-democracy protests since mid-March.
"Assad will stay president of Syria and his son will be president of Syria," one demonstrator told The Daily Star Sunday.
Protesters carried large Syrian flags, which they regularly lowered to the ground and kissed, the newspaper said. Protesters also carried posters of Assad.
"We have full freedom of expression" in Syria, said Syrian worker Sleiman Abdallah, 30, who said he didn't understand why people were demonstrating in his country.