Bloggers, media groups, and human-rights defenders opposed it, worried it was not well-thought-out - it was introduced last week - and could be loosely interpreted by the courts, which are seen as serving the interests of the authorities rather than observing the legal code.
Wikipedia on Tuesday closed its Russian-language website to protest the legislation.
Opposition groups, who have relied on the Internet to rally support, called it part of a broad assault on them.
Last month, the Duma passed a law drastically raising fines for protesters who violate the rules for holding a demonstration. On Wednesday, it advanced a bill that would make slander a criminal offense, with a penalty up to as much as the offender's annual income. Passage comes less than a year after then-President Dmitry A. Medvedev removed slander from the criminal code.
The Duma is expected to pass a law this week requiring nongovernmental organizations that do political work and get money from abroad to register as foreign agents.
"Many of us are now in danger," said Oleg Kozyrev, an influential blogger. "I see this Internet law as part of a package of repressive laws directed at the opposition and human rights and civil rights activists."
Under the legislation, he said, a commenter could post a link to child pornography on a blogger's page, for example, and the government would have the authority not only to remove the offending text but close the entire page. The page would remain closed while its owner tried to prove he was not responsible for the illegal reference.
The law is so broad that even vulgar language could be deemed an offense, said Alexander Morozov, a blogger and director of the Center for Media Research.
"Civic activists, journalists, and politicians all have grounds to be nervous," he said, observing that once authorities make their desires clear by charging someone, judges generally go along and find them guilty. "It could be applied to all kinds of statements on the Web. And we do not have acquittals in our courts."
This article contains information from Bloomberg News.