When a Pokémon with Earth Eater is hit by a Ground-type move, its HP is restored by one quarter of its maximum HP, and the move will have no effect on that Pokémon (if a Pokémon with this Ability already has full HP, the Ground-type move will do nothing).
Earth Eater will not activate if the Pokémon is protected from the Ground-type move.
Source
Bulbapedia gives no indication that the Pokemon is raised from the ground. The ability is simply a Ground type version of Water Absorb.
That means a Pokemon with Earth Eater would still be immune to Thousand Arrows, as it is still a Ground type move.