The average capacity of a single-input single-output (SISO) underwater wireless optical communication (UWOC) system with partially coherent Gaussian beams in a weak oceanic turbulence regime is investigated. An approximate analytical expression of scintillation index is derived mathematically to characterize the impact of oceanic turbulence on the propagation behavior of the partially coherent Gaussian beams. Then, the path loss caused by absorption and scattering in the ocean is numerically simulated with the Monte Carlo method. With consideration for absorption, scattering, and oceanic turbulence, the combined channel fading model is established, and the average capacity of the UWOC system (defined as the maximum mutual information between the input and output) is examined. Results show that the scintillations are reduced by decreases in propagation distance, the dissipation rate of mean-square temperature, and the ratio of the temperature and salinity contributions to the refractive index spectrum. Scintillations are also decreased by increases in source beam width, degree of partial coherence, and the dissipation rate of turbulent kinetic energy per unit mass of fluid. As a result, the average capacity of the UWOC system is enhanced. Moreover, the average capacity of the UWOC system can be promoted with the availability of channel state information at the receiver. This work will benefit the research and development of UWOC systems.