I guess a gamer can be a geek, and a geek can be a gamer. But one does not necessarily qualify as the other, except purely by the supposed association, which is just stereotyping. So, I say not either way is correct. Games and the people that play them, not to mention society's views on the definition of 'games', have changed so much that I believe the terms 'gamer' and 'geek' have gone so far as to not even associate one with the other automatically.
(Edit for clarity and spelling.)