The look in the eye phobia...?

I think the fact that you noticed this is reason enough for her to look away. If she noticed that you're constantly staring at her then there's a pretty good chance that she's at least a little creeped out by you. Eye contact only needs to be kept during a conversation. If you're just sitting there trying to stare into her eyes then it's no wonder she's looking away.

I used to keep eye contact at all times with anyone during a conversation because it's a sign of interest in what the person is saying. But I ran into an issue where really lonely/desperate/psychotic/etc. people would take that eye contact as something more, so I've sort of broken that habit. You can accomplish the same show of interest by nodding/saying 'yeah' when appropriate, asking questions and not fiddling with other things. Not saying to avoid eye contact completely, but it definitely should be broken every few moments.

But yeah, it really sounds like you're just being kind of creepy from what you've written. Staring at people is not cool. However, if someone catches you staring at them, you should make a funny face. Results won't be any worse than if they just caught you staring at them and can even turn out better because laughter is good for the soul. People who make funny faces are the best kind of people.