  It is already known that a simple nonlocal de Sitter gravity model, which we
denote as $\sqrt{dS}$ gravity, contains an exact vacuum cosmological solution
which mimics dark energy and dark matter and is in very good agreement with the
standard model of cosmology. This success of $\sqrt{dS}$ gravity motivated us
to investigate how it works at lower than cosmic scale -- galactic and the
solar system. This paper contains our investigation of the corresponding
Schwarzschild-de Sitter metric of the $\sqrt{dS}$ gravity model. To get exact
solution, it is necessary to solve the corresponding nonlinear differential
equation, what is a very complicated and difficult problem. What we obtained is
a solution of linearized equation, which is related to space metric far from
the massive body, where gravitational field is weak. The obtained approximate
solution is of particular interest for examining the possible role of non-local
de Sitter gravity $\sqrt{dS}$ in describing the effects in galactic dynamics
that are usually attributed to dark matter. The solution has been tested on the
Milky Way and the spiral galaxy M33 and is in good agreement with observational
measurements.
