I would argue that the main internal conflict faced by the father is actually the struggle to cling on to his humanity in a world where the few remaining humans seem to have abandoned any semblance of the humanity that we know and appreciate. The father, through trying to ensure the safety and protection of his son, is forced to kill and to be ruthless to guarantee survival. We see this when the father shoots the man that discovers them when they are hiding from the convoy of humans. We can likewise see it when they come across the old man who is travelling by himself. The father, out of grim experience, initially suspects the old man of being a decoy to entrap humans such as themselves, and then when they talk to him, the father threatens him, saying that if it is an ambush, the old man will be killed first.
It is the son that helps the father cling on to his humanity, both here and elsewhere in the book. It is the son that asks if they can spend the night with the old man, and give him some food:
Maybe we could give him something to eat.
He stood looking off down the road. Damn, he whispered. He looked down at the old man. Perhaps he'd turn into a god and they to trees. All right, he said.
Thus, arguably the central conflict that the father faces throughout the novel, both here and also in other locations, is to stay human and not become a brute, savage beast, killing and destroying others like so many other humans are depicted as doing in this novel. It is his son that acts as a check on his moral conscience, reminding him of his humanity.
In The Road, the main conflict that the man faces is how he will ensure the safety and survival of his son. It is implied that the man's wife, the boy's mother, commited suicide once they were forced to live on the road, so the man is the only person whom the boy has to look after him. Along the way, the harsh environmental conditions of the road along with the dangers posed by strangers become external conflicts for the man as he struggles to keep the boy safe and well fed. Internally, the man knows that he is growing weaker, and when illness takes over his body, he fears that he will die. The man tries to pass on good sense to his son so that he will be able to take care of himself. The man is able to resolve his conflicts by thinking carefully before acting, and he teaches his son how to remain compassionate even in dire times.