We perform global particle-in-cell simulations of pulsar magnetospheres, including pair production, ion extraction from the surface, frame-dragging corrections, and high-energy photon emission and propagation. In the case of oblique rotators, the effects of general relativity increase the fraction of the open field lines that support active pair discharge. We find that the plasma density and particle energy flux in the pulsar wind are highly non-uniform with latitude. We show that pulsar gamma-ray radiation is dominated by synchrotron emission, produced by particles that are energized by relativistic magnetic reconnection close to the Y-point and in the equatorial current sheet. In most cases, the calculated light curves contain two strong peaks, which is in general agreement with Fermi observations. The radiative efficiency decreases with increasing pulsar inclination and increasing efficiency of pair production in the current sheet, which explains the observed scatter in gamma-ray luminosity versus spin down energy losses. We demonstrate the formation of gamma-ray spectra and their spectral cutoff. Our findings lay the foundation for quantitative interpretation of Fermi observations of gamma-ray pulsars.