Sad to say, there is no such control in Java FX 2. However, Java FX 8 has one. Then again, if you want to maintain your rule that you must never force the user to update to the latest version just to be able to see a date picker, the only option is to make one.
Luckily, somebody else did the dirty work for us. Marko Jacob extended a more flexible version from Christian Schudt and it works flawlessly. Plus, using it is fairly easy too.
The downside to this is that you cannot import this control in the Java FX Scene Builder. You will have to add the DatePicker control programmatically.
Download his JavaFX Date Picker Release on GitHub.