The President can certainly sign an executive order saying pretty much whatever he wants to say. The President doesn't really have any explicit authority to issue executive orders, but historically executive orders are treated the same as Congressional legislation (although the courts can certainly overturn one).

That said, I would much prefer a proper legislative approach to an executive one. The Patent and Trademark Office is part of the Department of Commerce, so I don't think it would be frowned upon too much for the executive branch to enact some reform. Still, I think a legislative approach would be more robust and less controversial.