An equation to estimate the system size dependence of the self-diffusion coefficient of a tagged particle moving in a simple fluid is given using linear-response theory and linearized hydrodynamics. Estimates made by the equation are compared with the results of the molecular dynamics simulation for a hard-sphere fluid at two densities, rhosigma(3) approximately 0.88 and 0.47, where sigma is the hard-sphere diameter. Good agreement between theory and simulation is obtained at the higher density. At the lower density, the agreement becomes poorer, but it is improved by taking into account the diffusion effect of the tagged particle. The equation gives the same diffusion coefficient for the infinite system as that obtained by taking into account the long-time tail contribution of the velocity autocorrelation function [B. J. Alder, D. M. Gass, and T. E. Wainwright, J. Chem. Phys. 53, 3813 (1970)]. When the tagged particle has a larger mass than the fluid particles, the equation presented here gives the better estimates. It is confirmed by the molecular dynamics calculation.