Andrew, good point. This might do better if picking a date other than today.
Code:
Private Sub Document_ContentControlOnExit(ByVal CC As ContentControl, Cancel As Boolean)
If CC.Type = wdContentControlDate Then
If IsDate(CC.Range.Text) Then
If CC.DateDisplayFormat = "M/d/yyyy h:mm:ss am/pm" Then
CC.Range.Text = Format(CDate(CC.Range.Text), "M/d/yyyy") & " " & Time
End If
End If
End If
End Sub