Every year people come to the United States seeking protection because they have suffered persecution or fear that they will suffer persecution due to: Race, Religion, Nationality, Membership in a particular social group, or Political opinion.
You may only file an application for asylum if you are physically present in the United States, and you are not a U.S. citizen. More details on how to apply are here: https://www.uscis.gov/humanitarian/refugees-and-asylum/asylum
There are no asylum-related services available at U.S. Embassy in Tbilisi, or any other U.S. embassy.
The Government of Georgia registers asylum-seekers in Georgia. Additional information for asylum-seekers in Georgia can be found at United Nations High Commissioner for Refugees (UNHCR) https://help.UNHCR.org/Georgia.