  We report on a search for non-Newtonian forces that couple to mass, with a
characteristic scale of ${\sim}10~\mu$m, using an optically levitated
microsphere as a precision force sensor. A silica microsphere trapped in an
upward-propagating, single-beam, optical tweezer is utilized to probe for
interactions sourced from a nanofabricated attractor mass with a density
modulation brought into close proximity to the microsphere and driven along the
axis of periodic density in order to excite an oscillating response. We obtain
force sensitivity of ${\lesssim}10^{-16}~\rm{N}/\sqrt{\rm{Hz}}$. Separately
searching for attractive and repulsive forces results in the constraint on a
new Yukawa interaction of $|\alpha| \gtrsim 10^8$ for $\lambda > 10~\mu$m. This
is the first test of the inverse-square law using an optically levitated test
mass of dimensions comparable to $\lambda$, a complementary method subject to a
different set of systematic effects compared to more established techniques.
