  The Newtonian approximation with a nonvanishing nonlocal background field is
analyzed for the scalar-tensor nonlocal gravity and nonlocal Gauss-Bonnet
gravity. For these two theories, our calculations show that the Newtonian
gravitational constant $G$ is time-varying and $|\dot{G}/G|=\mathcal{O}(H_0)$
for the general case of cosmological background evolution, which is similar to
the results of the Deser-Woodard and Maggiore-Mancarella theories. Therefore,
observations about the orbit period of binary star (or star-planet) systems
could rule out these theories. One thing worth mentioning is that the nonlocal
Gauss-Bonnet gravity gives $\Psi=\Phi$ and a constant $G$ in the de Sitter
phase. Our results also highlight the uniqueness of the RT model [M. Maggiore,
\href{http://dx.doi.org/10.1103/PhysRevD.89.043008}{Phys. Rev. D {\bf 89},
043008 (2014)}], which is the only nonlocal gravity theory that can
successfully describe the gravitational phenomena from solar system to
cosmological scales for now.
