Why the Green Card does not help
The Green Card system is an agreement between the countries that joined it, under which each accepts the others' motor liability insurance. Georgia is not a member, so there is no arrangement for a foreign insurer to answer for damage caused on Georgian roads and no document a foreign insurer can issue that would satisfy the requirement.
This catches out drivers from countries where a Green Card covers most of the neighbourhood. It is worth checking rather than assuming: the card lists the countries it is valid in on its face, and Georgia is not among them.