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Protest leaders gave the government until 3 July to step down, vowing to gather again on that date.
The pro-EU demonstration came after the European Commission recommended on Friday that Georgia be denied EU candidate status.
The president made the comments in Brussels, where she said the Georgian government had blocked her from visiting European capitals, including Kyiv.
A second round between Nykhas Party head Alan Gagloyev and President Anatoly Bibilov is expected to be held in two weeks.
At a summit in Brussels, the leaders of Armenia and Azerbaijan also agreed to ‘work on the preparation of a future peace treaty’.
According to Ukraine, the smuggling routes were being established to import military, dual-purpose, and civilian goods that have been subject to Western export controls in response to Russia’s invasion of Ukraine.
The statement from President Anatoly Bibilov came as he fights for re-election and as South Ossetian soldiers in Russia’s military attack Ukraine.