The process γγ→ppK+K- and its intermediate processes are measured for the first time using a 980 fb-1 data sample collected with the Belle detector at the KEKB asymmetric-energy e+e- collider. The production of ppK+K- and a Λ(1520)0 (Λ(1520)0) signal in the pK- (pK+) invariant mass spectrum are clearly observed. However, no evidence for an exotic baryon near 1540 MeV/c2, denoted as Θ(1540)0 (Θ(1540)0) or Θ(1540)++ (Θ(1540) - ), is seen in the pK- (pK+) or pK+ (pK-) invariant mass spectra. Cross sections for γγ→ppK+K-, Λ(1520)0pK++c.c. and the products σ(γγ→Θ(1540)0pK++c.c.)B(Θ(1540)0→pK-) and σ(γγ→Θ(1540)++pK-+c.c.)B(Θ(1540)++→pK+) are measured. We also determine upper limits on the products of the χc0 and χc2 two-photon decay widths and their branching fractions to ppK+K- at the 90% credibility level.