Naturalism in literature is defined by a focus on realistic situations and scenarios, a lack of adventure or extraordinary events, the lives of simple people in a large world, and the darker or more tragic side of human living (Wikipedia).
Saul Bellow's works tend to focus on the intellectual pursuit of satisfaction, either in one's personal life or in regard to a specific quest. While he avoids many of the tropes of Romanticism -- a genre antithetical to Naturalism -- he is not so consumed with the suffering of living, instead showing the humanity of people in difficult situations. Some of his boos, such as Herzog, are overwhelmingly dark until the moment of epiphany; others, such as Henderson the Rain King, are deliberately light, almost joyous, in their treatment of human foible. Bellow is therefore not a strict naturalist; in fact, the Swedish Academy, in awarding Bellow the Nobel Prize for Literature 1976, "praised his early novels for breaking away from the harshness of naturalism and his later novels for their thought-provoking expansiveness" (faculty.atu.edu).
Strict Naturalism is harsh, as exemplified by J.D. Salinger and Emile Zola; Bellow shows compassion for his characters, and allows them a deeper humanity, and so as a whole his works cannot be placed into the Naturalist genre.