  It is well-known that there exists a close relation between a large $N$
matrix model and noncommutative (NC) field theory: The latter can be naturally
obtained from the former by expanding it around a specific background. Because
the matrix model can be a constructive formulation of string theory, this
relation suggests that NC field theory can also include quantum gravity. In
particular, the NC $U(1)$ gauge theory attracts much attention because its
low-energy effective action partially contains gravity where the metric is
determined by the $U(1)$ gauge field. Thus, the NC $U(1)$ gauge theory could be
a quantum theory of gravity. In this paper, we investigate the scenario by
calculating the scattering amplitude of massless test particles, and find that
the NC $U(1)$ gauge theory correctly reproduces the amplitude of the usual
graviton exchange if the noncommutativity that corresponds to the background of
the matrix model is appropriately averaged. Although this result partially
supports the relation between the NC $U(1)$ gauge theory and gravity, it is
desirable to find a mechanism by which such an average is naturally realized.
