Hi, C. You did not say whether the doors were stained and clear finished or painted. If they are painted, You can apply Minwax High Performance Wood Filler or Bondo to the gouged out areas just like you do with spackle on drywall. Sand down the filler, vacuum and tack cloth and repaint. If they are stained, you probably should call in a refinishing pro. Blending stain colors over fillers is a special art and takes a bit of experience and some special tools. You probably have a pretty big investment in your doors and want to keep them looking nice.
That said, now I am going to tell you to mess up the looks a little. A local window, screen and patio door supplier has aluminum expanded metal screens that go over the sliding screens on patio doors. I'm sure that you can find a place that carries them in your area, too, if you spend some time with the phone book. You will probably want to spray paint them with brass colored paint or something before you install them. Installation will require putting 4 screws (predrill)into your door to hold them on but 4 screw holes can be fixed a lot easier than redoing a whole bunch of new scratch marks in a year or so.
The only other solution that I have is to get rid of the dog and I'm sure that is unacceptable. You could get a cat but they can scratch worse than dogs. Ours loves to scratch all the furniture in the house and completely distains the scratching post that I built for her. Either that or it may just be her comment on some of my finishing jobs on the furniture.