That one chess lady (Ami help me out here) demonstrated that for strategy games, women can compete with men. Of course, she was a bit of an outlier.
The reason meat-space sports are segregated is because of the obvious physical advantages men have. Women and men have fairly equal scores on intelligence tests, so there does not appear to be the same obvious advantage for men in video games as in physical sports.
However, in an RTS game like Starcraft II, there does seem to be evidence that men would have a natural mechanical advantage. I went to a database and pulled this from the abstract of the first article I found on the topic:
Men are often found to have faster and less variable reaction times (RTs) than do women
Dykiert D, Der G, Starr J, Deary I. Sex differences in reaction time mean and intraindividual variability across the life span. Developmental Psychology [serial online]. September 2012;48(5):1262-1276
If I was the best girl in the world at a game, I'd wanna compete against men and prove how good I was. If I was only mediocre and it was a local tournament, frankly I'd rather play against chicks most of the time so I could do well. But I'm a "set the bar low" kinda guy.