I found the class ActionSheetDatePicker from the book Developing C# Apps for iPhone and iPad using MonoTouch iOS Apps Development for .NET Developers. It was a very nice and clean implementation that would allow for having an ActionSheet pop up with a UIDatePicker in an iOS application.

It did lack 2 things, though:

1. The ability to specify a starting default date in the UIDatePicker
2. The ability to get cleanly launched from a UITextField component (i.e. effectively replace the default keyboard of a text field with this date picker). The original code would leave the default keyboard up when getting called from a UITextField component.

I fixed those things, and you can find the results in this gist.