Despite the narrator's guilt over the situation, he is not to blame for his brother's death. He may have treated him with indifference and occasional malice, both of which are displayed by his words and actions in the story, but in the end, those words and actions were not the primary cause of Doodle's death.
We as readers know that his condition is what eventually led to his certain death, and that from the very beginning, Doodle was supposed to have died many times before. The fact that he had lasted as long as he had was a fact of sheer amazement.
In the end, the narrator cannot be held responsible for Doodle's death, but we can continue to hold him accountable for his harsh words and behaviors toward his now-deceased brother.
Now, of course, this is an opinion question, so the answer is up to you, as well as the evidence to support that answer.
In my opinion, however, the narrator is not at all responsible for Doodle's death. Although, of course, he feels terribly to blame.
Doodle was destined to die. To use this course of reasoning, you really have to argue that fate has a huge role in this story. This is evident in the scarlet ibis's presence, which was a foreshadowing of what was to come. Doodle was represented by this brilliant bird, and the bird died. Doodle was supposed to die.
The narrator was only 13. He was cruel, it's true, but he was also only 13. He was acting how most boys his age do when confronted with something uncomfortable, like a disabled brother. He was jealous of the attention Doodle got, and he was embarrassed by Doodle's difference. He was probably also disappointed that he didn't have a brother who could run, wrestle, and play with him as he'd probably originally hoped. There was a lot of emotion flooding through the narrator. There is really no way he could have known that Doodle was hurt as badly as he was. Plus, Doodle sustained a head injury and died so quickly that he would have died even if the narrator had run back sooner to help him.
And regardless of the narrator's motives for helping Doodle to walk, he did improve Doodle's quality of life for a short while, regardless of how frustrated Doodle sometimes got.