A coalition of opposition parties said at a joint briefing that protests would continue until the bill was officially rejected and those arrested were released.
At least 77 people were arrested Tuesday night as clashes broke out near Parliament. Fighting broke out again when the protesters returned on Wednesday evening.
Police fired tear gas and water cannons at protesters, who waved Georgian and EU flags and shouted “No to the Russian law”.
The episode has become a lightning rod for broader tensions in society and the ruling elite.
Irakli Garibashvili, prime minister and head of Georgian Dream, said the law will make the media more transparent and weed out foreign influence on domestic politics.
But Salome Surabichvili, the country’s president, has backed the protesters and threatened to veto the law if it were passed. Such a veto would not be final as Parliament can overrule the President.
The European Union delegation said it welcomed the decision to put the bill on hold.
Dmitry Peskov, Vladimir Putin’s spokesman, denied that Russia was involved in drafting the bill.
Georgia was annexed by Russia in the 19th century and regained its independence in 1991. The countries maintain close ties but have repeatedly come into conflict over Tbilisi’s efforts to join NATO and the EU.
Russia controls two breakaway regions that make up about 20 percent of the country and waged a war with Georgia in 2008, in part over its plans to join NATO.
Georgia applied for membership of the European Union last year, partly in response to the Russian invasion of Ukraine.
European leaders have said they must implement reforms before qualifying for official candidate status.