I think that in order to call yourself a Trekkie you should at the very least have seen most, if not all, of one of the shows. Just watching the movies alone does not make you a fan because the films are just a continuation of TOS and TNG. And the newest film is so far removed from everything before it that if you're just a fan of that alone you can only call yourself a ST09 fan, not a Trekkie or even a casual Trek fan.
Just my opinion.
I think we might be using the term "trekkie" too loosely. Perhaps what it really meant was someone who lives and breathes Star Trek. Someone who has gone to conventions in full Klingon/Borg/Starfleet (Or anything else) regalia. I mean if you like just one show, than you're a fan of that show but I wouldn't say you're a trekkie. I mean I've loved Star Trek my whole life but I don't consider myself a Trekkie. Perhaps it's that whole issue of what does it mean to be a casual fan vs. actual fanatic that we're debating here.