While hydrogen could be an important light alloying element in planetary iron cores, phase relations in the Fe-FeH system remain largely unknown at high pressures and temperatures (P-T). A speculative Fe-H₂ phase diagram has been proposed assuming continuous solid solution between Fe and FeH and eutectic melting between FeH and H₂. Recent studies revealed that stoichiometric FeH becomes non-magnetic above ~40 GPa, which might affect its melting behavior. Here we examined the melting curve of non-magnetic FeH between 43 and 152 GPa by a combination of laser-heated diamond-anvil cell (DAC) techniques and synchrotron X-ray diffraction (XRD) analyses. The melting temperature was determined by employing the appearance of additional hazy XRD signals upon quenching temperature as a melting criterion. We also performed thermodynamic modeling, which well reproduces the change in the curvature of FeH melting curve upon the loss of magnetism and extrapolates the experimental constraints to inner core pressures. The XRD data showed that non-magnetic FeH melts congruently at temperatures higher than the known eutectic melting curve for FeH_x (x > 1). Combined with the fact that the endmembers exhibit different crystal structures, these results indicate that Fe and non-magnetic FeH form a eutectic system. The dT/dP slope of the FeH melting curve is comparable to that for Fe, suggesting that the eutectic liquid composition of FeH_0.42 (Fe + 0.75 wt% H) previously estimated at ~40 GPa changes little with increasing pressure.