The merger of a neutron star (NS) binary may result in the formation of a long-lived, or indefinitely stable, millisecond magnetar remnant surrounded by a low-mass ejecta shell. A portion of the magnetar's prodigious rotational energy is deposited behind the ejecta in a pulsar wind nebula, powering luminous optical/X-ray emission for hours to days following the merger. Ions in the pulsar wind may also be accelerated to ultra-high energies, providing a coincident source of high energy cosmic rays and neutrinos. At early times, the cosmic rays experience strong synchrotron losses; however, after a day or so, pion production through photomeson interaction with thermal photons in the nebula comes to dominate, leading to efficient production of high-energy neutrinos. After roughly a week, the density of background photons decreases sufficiently for cosmic rays to escape the source without secondary production. These competing effects result in a neutrino light curve that peaks on a few day timescale near an energy of ∼ 10 18 eV. This signal may be detectable for individual mergers out to ∼ 10 (100) Mpc by current (next-generation) neutrino telescopes, providing clear evidence for a long-lived NS remnant, the presence of which may otherwise be challenging to identify from the gravitational waves alone. Under the optimistic assumption that a sizable fraction of NS mergers produce long-lived magnetars, the cumulative cosmological neutrino background is estimated to be ∼ 10 −9 − 10 −8 GeV cm −2 s −1 sr −1 for a NS merger rate of 10 −7 Mpc −3 yr −1 , overlapping with IceCube's current sensitivity and within the reach of next-generation neutrino telescopes.