Hundreds, both protesters and police officers, were injured in the clashes, some of them seriously, as demonstrators pushed against lines of riot police, threw bottles and stones, and grabbed riot shields, drawing a tough response.
Activists protested against the "Russian occupation" of Georgia, demanding the resignation of Parliament Speaker Irakli Kobakhidze, as well as the heads of Georgia's Interior Ministry and State Security Service.
"Russia is our enemy and occupier".
The small south Caucasus nation, a USA ally, lost a short war against Moscow in 2008.
President Vladimir Putin said he wanted Russian tourists visiting Georgia to be warned of the potential risks there.
Mr Gavrilov was taking part in the Interparliamentary Assembly on Orthodoxy (IAO) - a body set up by the Greek parliament in 1993 to foster relations between Christian Orthodox lawmakers. That angered some politicians and Georgians who want Russian Federation kept at arm's length. Currently on a visit to Minsk, Georgian President Salome Zourabichvili condemned Saakashvili's remarks, saying, "It's totally unacceptable for a citizen of another country [Saakashvili is now a Ukrainian citizen] to call, from overseas, on the police to disobey orders".
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Russian Deputy Foreign Minister Grigory Karasin said in a statement that Moscow was outraged by the actions of what he called radical Georgian political forces whom he accused of propagating anti-Russian sentiment.
Gavrilov told a Moscow news conference on Friday he believed the protests had been pre-planned.
Rustavi 2 cited former Georgian Defense Minister and leader of the Victorious Georgia movement, Irakli Okruashvili, as saying: "either the police will let the demonstrators through, or the people will use their constitutional right".
The opposition, which has seized on the furore to press much wider and unrelated demands, called on people to take to the streets again on Friday evening at 1900 local time (1500 GMT). Saakashvili, who has embarked on a political career in Ukraine, has said the Georgian accusations against him are motivated by revenge and has accused Ivanishvili, who made his billions in Russian Federation, of allowing increasing Kremlin influence in Georgia.
Kremlin spokesman Dmitry Peskov said the safety of Gavrilov and other members of the Russian delegation had been endangered.
A Reuters witness said that Tbilisi's main thoroughfare, Rustaveli Avenue, which runs in front of parliament, was closed to traffic on Friday and that the building itself was heavily guarded by police.
"It is unacceptable that a representative of the occupier country chairs a forum in the Georgian parliament", the billionaire tycoon said.
Georgia and its Soviet-era master Russian Federation have always been at loggerheads over Tbilisi's bid to join the European Union and North Atlantic Treaty Organisation with the spiralling confrontation culminating in a full-out war on August 8, 2008.