No, he's mortal! He just gets help from gods! He dies like fifteen times throughout the duration of the God of War trilogy! He just gets fished out of the river styx by various people...
... ok he WAS a god for a little while, but i'm talking the God of War 3, eye poking out, head ripping off, leg shearing Kratos who WAS mortal!
You have to admit, he has no super powers, he just kills people with super powers, steals their body parts that contain said powers and uses those himself. If he were in left for dead he'd be running around with a tanks muscles strapped across his arms, kicking ass.