We have performed the first direct measurement of the time-integrated flavor untagged charge asymmetry in semileptonic ${B}_{s}^{0}$ decays ${A}_{\mathrm{SL}}^{s,\mathrm{unt}}$ by comparing the decay rate of ${B}_{s}^{0}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{D}_{s}^{\ensuremath{-}}\ensuremath{\nu}X$, where ${D}_{s}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\phi}{\ensuremath{\pi}}^{\ensuremath{-}}$ and $\ensuremath{\phi}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}$, with the charge-conjugate ${\overline{B}}_{s}^{0}$ decay rate. This sample was selected from $1.3\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of data collected by the D0 experiment in run II of the Fermilab Tevatron collider. We obtain ${A}_{\mathrm{SL}}^{s,\mathrm{unt}}=[1.23\ifmmode\pm\else\textpm\fi{}0.97(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.17(\mathrm{syst})]\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}$. Assuming that $\ensuremath{\Delta}{m}_{s}/{\overline{\ensuremath{\Gamma}}}_{s}\ensuremath{\gg}1$, this result can be translated into a measurement of the $CP$-violating phase in ${B}_{s}^{0}$ mixing: $\ensuremath{\Delta}{\ensuremath{\Gamma}}_{s}/\ensuremath{\Delta}{m}_{s}\mathrm{tan}{\ensuremath{\phi}}_{s}=\phantom{\rule{0ex}{0ex}}[2.45\ifmmode\pm\else\textpm\fi{}1.93(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.35(\mathrm{syst})]\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}$.