We have performed a search for $CP$ violation in a sample of ${B}_{s}^{0}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{D}_{s}^{\ensuremath{-}}X$ decays corresponding to $5\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of proton-antiproton collisions collected by the D0 detector in Run II at the Fermilab Tevatron Collider. New physics in ${B}_{s}^{0}$ mixing could contribute a significant $CP$ violating weak phase, which would be observed as a difference in the decay-time distribution for ${B}_{s}^{0}\ensuremath{\rightarrow}{\overline{B}}_{s}^{0}$ oscillated states versus that for ${\overline{B}}_{s}^{0}\ensuremath{\rightarrow}{B}_{s}^{0}$. A fit to the decay-time distributions of the ${B}_{s}^{0}/{\overline{B}}_{s}^{0}$ candidates yields the flavor-specific asymmetry ${a}_{\mathrm{fs}}^{s}=[\ensuremath{-}1.7\ifmmode\pm\else\textpm\fi{}9.1(\mathrm{stat}{)}_{\ensuremath{-}1.5}^{+1.4}(\mathrm{syst})]\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$, which excludes $CP$ violation due to new physics within the experimental sensitivity.