The RadDatePicker's textbox (which is a RadDateInput control) shows its invalid style only if the user enters something, which cannot be parsed to a valid date. I am afraid that integration between RadDateInput and RequiredFieldValidator, which could exhibit the desired behavior, does not exist.
However, you can use a CustomValidator and trigger the picker's dateinput invalid style manually. Here is a similar example for RadTextBox that will give you an idea what needs to be done:
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