Government watchdogs have expressed concern that the law could be abused and harm free speech.
The bill was passed in its first reading and requires two more before becoming law, said a statement from Justice Minister Ayelet Shaked, one of its sponsors.
Public Security Minister Gilad Erdan, who also sponsored the legislation, called it "essential to provide the tools to act immediately to remove content that can lead to terrorist acts and murder".
It would also apply to major sites such as YouTube and Google.
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Material would be removed in cases where it poses "a real risk to the security of a person, the public or the state", Shaked has said.
Israel has previously held discussions with Facebook officials to stop what it calls online incitement.
In September, Shaked said that the social network giant had removed 95 per cent of the posts Israel had referred to it.
But the possibility for error was seen in September, when Facebook apologised after temporarily disabling accounts linked to two Palestinian news sites critical of Israel.
The move drew concern over the potential for online censorship.
The Israel Democracy Institute think tank said the bill was "unprecedented" in its current form when compared to similar legislation in other countries.
It said it would be difficult to enforce and would "facilitate a disproportionate amount of censorship".
In addition to the bill, Israeli and American victims of Palestinian attacks filed a USD 1 billion lawsuit against Facebook in July over allegations it was used by the Palestinian Islamist movement Hamas to organise violence.