This question stifles many an investigator, and the official stance of Facebook is that it does not report to any Facebook user who's visiting the site. Furthermore, it also ensures that no third party Apps can do so either. So the good news is no, your ex-girlfriend can’t see how many times you’re checking out her page to see if you have another shot.
However, there is a feature on Facebook called “People You May Know”, which shows up in the same place on your Profile where “Friend Requests” is located. That could very well act in a similar way to inform someone you’re looking at their profile when they are not friends with you yet. If you don’t see it near the top-right corner of your page right away when you login, try clicking on News Feed on the left, or Home in the top-right corner.