A direct numerical simulation (DNS) code is developed to simulate immiscible two-phase flows based on the recently developed discrete unified gas-kinetic scheme (DUGKS). This scheme simulates hydrodynamic equations of the quasi-incompressible Cahn-Hilliard-Navier-Stokes system by the use of two mesoscopic distributions and the proper design of their equilibrium distributions and source terms. Several immiscible two-phase flows are used to validate the scheme in both 2D and 3D, including a stationary droplet in 2D and 3D, the Rayleigh-Taylor flows, and two-phase homogeneous isotropic decaying turbulence. The results obtained by DUGKS are compared carefully to these from the literature and the ARCHER code, i.e., a Coupled Level Set-Volume of Fluid (CLSVOF) method. The comparisons indicate that DUGKS is a promising scheme for direct numerical simulations of immiscible two-phase flows.