The time-integrated CP asymmetry in the Cabibbo-suppressed decay D^{0}→K^{-}K^{+} is measured using proton-proton collision data, corresponding to an integrated luminosity of 5.7 fb^{-1} collected at a center-of-mass energy of 13 TeV with the LHCb detector. The D^{0} mesons are required to originate from promptly produced D^{*+}→D^{0}π^{+} decays, and the charge of the companion pion is used to determine the flavor of the charm meson at production. The time-integrated CP asymmetry is measured to be A_{CP}(K^{-}K^{+})=[6.8±5.4±1.6]×10^{-4} where the first uncertainty is statistical and the second systematic. The direct CP asymmetries in D^{0}→K^{-}K^{+} and D^{0}→π^{-}π^{+} decays, a_{K^{-}K^{+}}^{d} and a_{π^{-}π^{+}}^{d}, are derived by combining A_{CP}(K^{-}K^{+}) with the time-integrated CP asymmetry difference, ΔA_{CP}=A_{CP}(K^{-}K^{+})-A_{CP}(π^{-}π^{+}), and other inputs, giving a_{K^{-}K^{+}}^{d}=(7.7±5.7)×10^{-4},a_{π^{-}π^{+}}^{d}=(23.2±6.1)×10^{-4},with a correlation coefficient corresponding to ρ=0.88. The compatibility of these results with CP symmetry is 1.4 and 3.8 standard deviations for D^{0}→K^{-}K^{+} and D^{0}→π^{-}π^{+} decays, respectively. This is the first evidence for direct CP violation in a specific D^{0} decay.