There doesn’t seem to be a way to do this in Axure as “pre-built” HTML forms like Date and Number don’t seem to fire any meaningful events, nor can you get their values. So, a user can select a date in your field using the pre-built form, like the popup calendar, but no event is fired, like Text Changed or Key Up and if you try to refer to “text on widget MyDateField” or get [[LVAR1.text]] nothing is returned. Worse, if the user tries to manually enter or edit a date, it won’t work in most browsers. As a consequence, if you need to apply logic or in any way use a date in a field, you pretty much have to use a regular text field or sets of fields (e.g., Month, Day, Year) and validate/format the input.
See this thread for a good hand-built custom date picker: