Answers

It looks like the dog did a pretty good job on the door. Since the door is painted, you could sand the area down with some 150 grit sand paper. After it is sanded then you can apply some High performance wood filler or car bondo.

Apply a thin layer over the damaged area and allow it dry ( a few minutes or more depending on the amount of hardener you use). After it is dry, sand untill smooth and then apply a second layer if you need to and then sand smooth. After the area is the way you want it, prime it and paint it. 

Consider, a new dog

There are a few options depending on the circumstances, if the dog has scratched the slab and depending on material, you'll need an entirely new slab. You have the option of painting over the scratch, but the odds of that peeling into the future are high. Our doors are steel and fibreglass, fibreglass would require an entirely new door slab to be installed if you wish to rid the problem for good. To surely rid the problem, the dog or puppy should be trained before repair unless you want to experience a fruitness endeavour. 

When dealing with a wooden door, their are a few DIY solutions like sanding, filling sanding again and painting/staining over. 

It looks like the pup teaches classes on how to be a beaver 101. Sanding, filling and matching the colour back would be the best option in this circumstance, or simply an entirely new replacement. 

I would concur with the above answers.  If you are able to fill with wood putty and paint that would prove to be the most cost effective option.  If you would like to go further, determine the brand of the door and I'm sure you could find a replacement sash.  I am like you, I love my dog like my children, continue to be patient and show your bowser the love he deserves.

As with the previous comments, you can use sandpaper to smooth out some of the damage, but be sure to rub the scratches along the grain of the wood. Next, apply mineral spirits (these are solvents made from paint thinner and petroleum) over the sanded area. Spread on wood filler with a Spackle knife to the damaged parts and allow it to dry as per the filler’s directions. After the wood filler has dried, lightly sand the area once again and brush away any dust before priming and painting your door.

While the image shows damage to the inside of your door, your pet may try scratching on the outside of door and its weather stripping in an attempt to be let inside. In order to prevent future damage to that part of your door, you can add a piece of vinyl lattice that matches the color of your doorframe. Cut the lattice to the length of the doorjamb. Place it over your foam weather stripping. Check to see that your door continues to close properly. Use finishing nails every foot to tack the vinyl lattice securely in place. A small amount of spackling will cover up the nail heads. Now the lattice will protect your weather stripping from further damage.

The right type of door for your home can make a huge difference. We install ProVia doors and they offer DuraGuard Series Storm Doors, which are ideal for keeping pets safely inside with their non-removable stainless steel screening.  

Best of luck with your dog and your door repair!