The dog Spitz dies at the very end of Chapter 3. That is the chapter that is entitled "The Dominant Primordial Beast."
Spitz dies after a long chase and a fight with Buck. The two of them have been contending for power within the pack all through this chapter. Buck has been undermining Spitz's authority for a long time and he has finally taken the opportunity to pick a fight to the death with Spitz. They fight for a long time, and Buck finally gets Spitz down. At that point, Spitz dies just the way he killed Curly earlier in the book.