  Effects of the non-Newtonian gravity on properties of finite nuclei are
studied by consistently incorporating both the direct and exchange contribution
of the Yukawa potential in the Hartree-Fock approach using a well-tested Skyrme
force for the strong interaction. It is shown for the first time that the
strength of the Yukawa term in the non-Newtonian gravity is limited to
$\log(|\alpha|)<1.75/[\lambda(\rm fm)]^{0.54} + 33.6$ within the length scale
of $\lambda=1-10$ fm in order for the calculated properties of finite nuclei
not to be in conflict with accurate experimental data available.
