No, you cannot.  The documentation is pretty clear on this:
This means, for example, that you cannot set the default for a date
  column to be the value of a function such as NOW() or CURRENT_DATE.
  The exception is that you can specify CURRENT_TIMESTAMP as the default
  for TIMESTAMP and DATETIME columns. See Section 12.3.5, “Automatic
  Initialization and Updating for TIMESTAMP and DATETIME”.
You can do one of the following:
- Set up a column with a default value for the DATETIME.  Create view that extracts the date as a separate column.
- Create an inserttrigger to set thedatecolumn.