We study the process γγ→J/ψω using a data sample of 519.2 fb−1 recorded by the BABAR detector at SLAC at the PEP-II asymmetric-energy e+e− collider at center-of-mass energies near the Υ(nS) (n=2,3,4) resonances. We confirm the existence of the charmoniumlike resonance X(3915) decaying to J/ψω with a significance of 7.6 standard deviations, including systematic uncertainties, and measure its mass (3919.4±2.2±1.6) MeV/c2 and width (13±6±3) MeV, where the first uncertainty is statistical and the second systematic. A spin-parity analysis supports the assignment JP=0+ and therefore the identification of the signal as due to the χc0(2P) resonance. In this hypothesis we determine the product between the two-photon width and the final state bran...