A search is presented for lepton-flavor violating decays of the Higgs boson to $\ensuremath{\mu}\ensuremath{\tau}$ and $e\ensuremath{\tau}$. The dataset corresponds to an integrated luminosity of $137\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ collected at the LHC in proton-proton collisions at a center-of-mass energy of 13 TeV. No significant excess has been found, and the results are interpreted in terms of upper limits on lepton-flavor violating branching fractions of the Higgs boson. The observed (expected) upper limits on the branching fractions are, respectively, $\mathcal{B}(H\ensuremath{\rightarrow}\ensuremath{\mu}\ensuremath{\tau})<0.15(0.15)%$ and $\mathcal{B}(H\ensuremath{\rightarrow}e\ensuremath{\tau})<0.22(0.16)%$ at 95% confidence level.