The process gamma gamma -> p (p) over barK(+)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 p (p) over barK(+)K(-) and a Lambda(1520)(0) ((Lambda) over bar (1520)(0)) signal in the pK(-) ((p) over barK(+)) invariant mass spectrum are clearly observed. However, no evidence for an exotic baryon near 1540 MeV/c(2), denoted as Theta(1540)(0) ((Theta) over bar (1540)(0)) or Theta(1540)(++) (Theta(1540)(--)), is seen in the pK(-) ((p) over barK(+)) or pK(+) ((p) over barK(-)) invariant mass spectra. Cross sections for gamma gamma -> p (p) over barK(+)K(-), Lambda(1520)(0)(p) over barK(+) + c.c. and the products sigma(gamma gamma -> Theta(1540)(0)(p) over barK(+) + c.c.)B(Theta(1540)(0) -> pK(-)) and sigma(gamma gamma -> Theta(1540)(++)(p) over barK(-) + c.c.)B(Theta(1540)(++) -> pK(+)) are measured. We also determine upper limits on the products of the chi(c0) and chi(c2) two-photon decay widths and their branching fractions to p (p) over barK(+)K(-) at the 90% credibility level.