if you already have a cx600 don't bother upgrading it until you need to, it's fine for you're build and still has a bit of room for other upgrades besides another gpu. Now down the road if you decide you want to xfire or get a really expensive card like a 7990 then id reccomend upgrading the psu.
yep you have a quality psu, thats more then enough to run one 7970, might run into a issue if you go with a 2-3way xfire depending on what else you plug into your computer.
My CPU is the fx 8350 and my mobo is the asrock 970 extreme3. With 8 gigs of ram. So could this PSU handle this GPU with these system specs?
your PSU can handle it perfectly fine. the PSU cannot however handle a Crossfire mode of this GPU. as it is only a 46A on the 12V rail, 2x 7970s need more.
if you already have a cx600 don't bother upgrading it until you need to, it's fine for you're build and still has a bit of room for other upgrades besides another gpu. Now down the road if you decide you want to xfire or get a really expensive card like a 7990 then id reccomend upgrading the psu.