The four main opposition groups have rejected Kavelashvili and boycotted parliament.
It is as yet unclear how the stand-off will be resolved.
Protesters, waving Georgian and EU flags, formed a human chain that spanned kilometres on Saturday.
“I am out in the street together with my whole family trying somehow to tear out this small country out of the claws of the Russian empire,” one protester told the Associated Press.
Georgian Dream has become increasingly authoritarian in recent years, passing Russian-style laws targeting media and non-government groups who receive foreign funding, and the LGBT community.
It refused to join Western sanctions on Russia after the full-scale invasion of Ukraine, and called the West the “global war party”, making a mockery of its stated aim of joining the EU and Nato.
An overwhelming majority of Georgians back the country’s path to the EU and it is part of the constitution.
But in November, the country’s ruling party said the government would not seek EU accession talks until 2028.
The announcement sparked days of protests, and riot police used tear gas and water cannon against protesters, who fought back by throwing fireworks and stones.
The US this week imposed sanctions on Georgia’s former prime minister and billionaire founder of Georgian Dream, Bidzina Ivanishvili.
Georgia is a parliamentary democracy with the president the head of state, and the prime minister the head of parliament.
The current president, Zourabichvili, has denounced Kavelashvili’s election – which was under an electoral college system in which he was the only candidate – as a travesty.
When Zourabichvili became president in 2018 she was endorsed by Georgian Dream, but she has since condemned their contested election victory in late October as a “Russian special operation” and backed nightly pro-EU protests outside parliament.
Zourabichvili has vowed not to step down on Sunday.
The government says if she refuses to leave office she will be committing a crime.