why does candy say that he should have shot his own dog?

Candy realises that he should have taken responsibility for his friend and made the humane gesture to shoot the dog himself. He regrets the fact that the dog's last moments were with Carlson, who did not care for him, rather than together.

He feels that he let his friend down, and that perhaps he should have made the decision earlier. The dog was old, blind and had no teeth. Its quality of life had been limited for some time.