I looked for F. Walters from Georgia and could only find one Franklin Walters that would fit, the other Franklin was listed as born in 1842. There was also a Fred M. Walters. And that's the total.
You'd need to see his service record file to try to identify him. See the advertisement above for a service providing these records.
George Martin
Franklin Walters
Residence Hart County GA;
Enlisted on 3/4/1862 as a Private.
On 3/4/1862 he mustered into "H" Co. GA 15th Infantry
(date and method of discharge not given)
He was listed as:
* POW 7/3/1863 Gettysburg, PA
* Released 6/7/1865 Fort Delaware, DE