I reside in Texas but I work remotely as an independent contractor in Georgia. Do I file a Georgia State income tax return?

No.

All your income is considered as sourced in Texas,

Thank you!

Now I only have one problem. I paid estimated taxes to Georgia. 

To retrieve that money, should I file a Georgia state income tax return?

You should file a non-resident tax return for Georgia, reporting 0 Georgia income to claim a refund of those estimated taxes paid.