A numerical method for solving Navier-Stokes equations and combined with a phase-field interface model is applied to flow problems of motion of an incompressible isothermal two-phase fluid with a high density ratio on a solid surface. Based on the free-energy theory, a fluid interface is described as a finite volumetric zone across which the physical properties vary continuously. The wettability of a solid surface is taken into account through a simple boundary condition derived from the increase in free energy on the surface. The phase-field approach simplifies the capture of motions of a fluid interface on a surface (contact line). The major findings from the simulations are as follows:(1) the contact-line motions of the liquid column under gravity are well predicted in comparison with the available data;(2) the static contact angle is flexibly controlled by a parameter of the wetting potential of the surface;(3) the capillary force is evaluated appropriately;(4) the acceleration of the two-phase flow in a channel with a local hydrophilic surface is predicted and observed to be in qualitative agreement with the experimental data; and (5) the displacement and breakup of a single drop on a flat solid wall are well predicted qualitatively. These results prove that the phase-field method can be employed for simulating air-water flows on a surface with heterogeneous wettability.