Georgia does not recognize a same sex marriage as a legal civil contract. Even if you were married in another state that does permit and recognize same sex marriage as a legal civil contract. The state of Georgia also does not allow same sex divorce to occur within the Georgia court system.
It may allow a gay marriage from another state to be annulled, but the circumstances are very specific. This would only occur if one party could show or prove that great harm would occur if the marriage were allowed to continue. In Georgia, a same sex marriage and a same sex divorce would need to be handled by lawyers. The process for a same sex divorce or the dissolution of a same sex marriage would not be quick. In fact, it could take the court years to grant an annulment.
All couples in Georgia, who have had a same sex marriage would have done such in another state. It is advisable that for those couples who want a same sex divorce that they begin that process in the state where their same sex marriage took place. Some states, such as California, allows same sex couples who were wed in California and now live in another state to file for a same sex divorce in California. California also accepts faxed divorce petitions so said couple would not have to be in California to file for divorce. A good source to complete your divorce petition is completecase.com. The site makes it simple, cheap and easy to complete your divorce petition. There is a pre-qualification quiz that will tell you if you qualify. Then just answer the questions about property, debts, assets, and children. The system will use your answers to fill in a court approved divorce petition.
Most states that allow same sex marriage have some arrangement for dealing with the dissolution of a marriage specifically because not all states recognize same sex unions. Start with the state where you were married. In the meantime, you can check with completecase.com and see if you qualify to use the system. There are resources there that may also help you. It is hoped that in time all 50 states within the United States will recognize same sex marriage and offer same sex divorce for married couples.