Using Safari on an iPad, or the iOS Simulator in XCode, simply go to the GWT showcase for the DatePicker:
Most of the time, when you tap a date, it's selected. However, sometimes the DatePicker is left in an in-between state where the date is highlighted yellow (as if the user had hovered over the date with a mouse) but it is not selected. Tap again, and if you're lucky, it's finally selected.
The current release is GWT 2.5, and I'm not sure when this problem appeared...