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It is already known that a simple nonlocal de Sitter gravity model, which we denote as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msqrt><mrow><mi>d</mi><mi>S</mi></mrow></msqrt></semantics></math></inline-formula> gravity, contains an exact vacuum cosmological solution that mimics dark energy and dark matter and is in very good agreement with the standard model of cosmology. This success of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msqrt><mrow><mi>d</mi><mi>S</mi></mrow></msqrt></semantics></math></inline-formula> gravity motivated us to investigate how it works at a lower-than-cosmic scale—galactic and the solar system. This paper contains our investigation of the corresponding Schwarzschild–de Sitter metric of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msqrt><mrow><mi>d</mi><mi>S</mi></mrow></msqrt></semantics></math></inline-formula> gravity model. To obtain an exact solution, it is necessary to solve the corresponding nonlinear differential equation, which is a very complicated and difficult problem. What we obtained is a solution to a linearized equation, which is related to space metrics far from the massive body, where the gravitational field is weak. The obtained approximate solution is of particular interest for examining the possible role of nonlocal de Sitter gravity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msqrt><mrow><mi>d</mi><mi>S</mi></mrow></msqrt></semantics></math></inline-formula> in describing the effects in galactic dynamics that are usually attributed to dark matter. This solution was tested on the Milky Way and the spiral galaxy M33 and is in good agreement with observational measurements.