Yes, you can use a grey card to set your white balance....the reason is that it's color neutral. You could essentially use anything that is color neutral.
There is a proper way to set your custom WB though, and I think maybe that's where you messed up. You need to put the card in the same light that you'll be shooting in (or that you'll have in your photo). Then take a photo of the card. Then go into the menu and set the custom WB using that image.
In this example, you have a lot of light coming in via some windows. It's uncertain what type/color of light is illuminating the dog. I'd guess that when you took the sample photo of the grey card, you didn't have it in the same light that is mostly influecnecing your image...so the results are not what you're expecting.
To this same point, if the dog is being lit by different light than what is in the room behind him, then you'll get a color cast to that room (or if the room was accurate, the dog would have a cast). In other words, it looks like you're trying to take a shot that has two very different light sources that don't have matching color temps...and that is always a recipe for disaster.
Lastly, make sure you are using the 'custom' (PRE) white balance setting, not the 'manual' WB setting. The manual setting is where you can dial in the specific color temperature. It's really only useful if you have a color meter (few do).