A search for the very rare D^{0}→μ^{+}μ^{-} decay is performed using data collected by the LHCb experiment in proton-proton collisions at sqrt[s]=7, 8, and 13 TeV, corresponding to an integrated luminosity of 9 fb^{-1}. The search is optimized for D^{0} mesons from D^{*+}→D^{0}π^{+} decays but is also sensitive to D^{0} mesons from other sources. No evidence for an excess of events over the expected background is observed. An upper limit on the branching fraction of this decay is set at B(D^{0}→μ^{+}μ^{-})<3.1×10^{-9} at a 90% C.L. This represents the world's most stringent limit, constraining models of physics beyond the standard model.