Is the narrator responsible for Doodle's death?

Yes, although he clearly did not mean for Doodle to die. The narrator was upset that Doodle had not progressed physically as far as they had wanted and he was running too fast for Doodle to keep up almost to punish Doodle for his failure to progress. The brother knew Doodle could not over exert himself, but he did not think about that. He should have been more sensitive to Doodles' condition than to his own preferences for a healthier brother.