seriously imho, it has nothing to do with the dog.
I think your wife is not really ready to become a mom. Because you can have a dog and have a kid together as well. Try go check out lots of videos in Youtube where a couple with existing dog welcome a child and the dog even become good buddy of the child and very protective of the child too. (maybe you can consider showing such video to your wife too)...
but take a way the dog or put aside the dog, i think without the dog too your wife most likely will give you the same answer that she's not ready to become a mom...

Taking care of a child is way way way way way more complicated and harder than taking care of a dog... i'm speaking from my own experience...i have dog...i also have kids... 
good luck buddy...

on a separate note....you love your wife and you made a vow to marry her... just because she don't want to have kids you want to divorce her? you love your wife or you love kids? again, just my own opinion, I love my wife first and whatever after that, both should discuss and come to mutual agreement together. In this case, if my wife is not ready to have kids, i shall stick with her decision as well until she is ready...even if it means there is a possibility we won't have kids...