XMM-Newton and Chandra Observations of the Candidate Fermi-LAT Pulsar 4FGL J1015.5-6030

4FGL J1015.5-6030 is an unidentified Fermi-LAT source hosting a bright, extended X-ray source whose X-ray spectrum is consistent with that of a young pulsar, yet no pulsations have been found. Here we report on XMM-Newton timing and Chandra imaging observations of the X-ray counterpart of 4FGL J1015.5-6030. We find no significant periodicity from the source and place a 3σ upper limit on its pulsed fraction of 34%. The Chandra observations resolve the point source from the extended emission. We find that the point source's spectrum is well fit by a blackbody model, with temperature kT = 0.205 ± 0.009 keV, plus a weak power-law component, which is consistent with a thermally emitting neutron star with a magnetospheric component. The extended emission spans angular scales of a few arcseconds up to about 30″ from the point source and its spectrum is well fit by a power-law model with a photon index Γ = 1.70 ± 0.05. The extended emission's spectrum and 0.5-10 keV luminosity of 4 × 10³² erg s^-1 (at a plausible distance of 2 kpc) are consistent with that of a pulsar wind nebula. Based on a comparison to other GeV and X-ray pulsars, we find that this putative pulsar is likely a middle-aged (i.e., τ ~ 0.1-1 Myr) radio-quiet pulsar with $\dot{E}\sim {10}^{34}\mbox{--}{10}^{35}$ erg s^-1.