For interface-tracking simulation of two-phase flows with a high density ratio, a computational method, NS-PFM, combining Navier-Stokes (NS) equations with a phase-field model (PFM) is proposed in this study. In accordance with the free energy theory, PFM describes an interface as a finite volumetric zone across which physical properties vary continuously. Surface tension is defined as an excessive free energy per unit area induced by the density gradient in the interface zone. The Cahn-Hilliard (CH) equation is used for predicting interface configuration. PFM therefore does not require any elaborating interface-tracking algorithms for predicting interface motion. The proposed NS-PFM is applied to several problems of incompressible, isothermal two-phase flow with the same density ratio as that of an air-water system. Numerical simulations demonstrate that, (1) the volume flux driven by local chemical potential gradient in the CH equation plays an important role in interfacial advection and reconstruction, (2) the NS-PFM gives good predictions for pressure increase inside a bubble caused by the surface tension, (3) predicted collapse of two-dimensional liquid column in a gas under gravity agrees well with available data, and (4) coalescence of free-fall single drop into a liquid film is successfully simulated in three dimensions, without using conventional interface-capturing/tracking techniques.