No.
You can have an automatic date for the last date printed, the last date saved, the current date, or the date created. To have the saved date work, you must save the file, update the field, and then save again.
Using Date Fields in Microsoft Word

You could conceivably have a macro, but remember that an action that Word considers a change may not be one you consider a change. For instance, if I type a space and then backspace to delete it, Word considers that a change.