  In this paper, we discuss the normality of the toric rings of stable set
polytopes, and the set of generators and Gr\"obner bases of toric ideals of
stable set polytopes by using the results on that of edge polytopes of finite
nonsimple graphs. In particular, for a graph of stability number two, we give a
graph theoretical characterization of the set of generators of the toric ideal
of the stable set polytope, and a criterion to check whether the toric ring of
the stable set polytope is normal or not. One of the application of the results
is an infinite family of stable set polytopes whose toric ideal is generated by
quadratic binomials and has no quadratic Gr\"obner bases.
