At the end of the novel, Billy has lost almost everything in his life. He lost the female wolf, lost his family and home, and lost his brother to death. Even his attempt to bring his brother back for a proper burial is thwarted with extreme cruelty. One thing that Billy cannot seem to find is a proper place for himself; he is drawn to the road despite his attempts to find his place in the world, and despite advice given to the contrary.
By the time he meets the stray dog, which only wants human companionship, he is bitter and disillusioned about his life, and he wants to wallow in his anger alone. He drives the dog away -- a mistake, as he could have begun to heal from his ordeals through this interaction. Instead, he doesn't think about the importance of companionship until the very end, but it is too late.
He called and called. Standing in that inexplicable darkness. Where there was no sound anywhere save only the wind. After a while he sat in the road. He took off his hat and placed it on the tarmac before him and he bowed his head and held his face in his hands and wept. He sat there for a long time and after a while the east did gray and after a while the right and godmade sun did rise, once again, for all and without distinction.
(McCarthy, The Crossing, Google Books)
The ending shows that Billy has been more-or-less broken by his experiences. He cannot get ahead through his own actions, and he can't seem to affect the world around him in a positive way. Everything he tries ends in failure. The last line, referencing the sun rising "for all and without distinction," shows that the world will continue regardless of his actions; Billy is ultimately meaningless in the larger scheme of things, as the sun will rise on anyone, anywhere. He is not special or important; he is simply a person like any other, but with the burden of extreme bad luck, some bad choices, and ultimately a worldview of bleak determinism. Billy is, for better or for worse, doomed to failure.