Efficient and robust numerical methods for incompressible two-phase flow are essential in various engineering applications. A common challenge in these methods arises from the absence of a time evolution equation for pressure in the incompressible Navier-Stokes equation. This challenge is typically addressed by solving the costly Poisson's equation. Over the years, various alternative methods have been suggested, including the artificial compressibility method, the lattice Boltzmann method, and more recently, weakly compressible (WC) methods based on the thermodynamic theory of incompressible flows. The relaxed divergence-free velocity field condition in WC methods allows for the development of fully-explicit algorithms for incompressible flows, known for their excellent scalability to large-scale problems. In the present work, a fully-explicit weakly compressible method is developed for solving incompressible two-phase flows. Here, the general pressure equation [1] is coupled with the momentum and conservative level set [2] equations to model the two-phase flow. The governing equations are solved in a closely coupled manner to capture the correct jump conditions across the fluid-fluid interface. A novel Riemann solver capable of accurately capturing the jump conditions, verified using the generalized Riemann invariant analysis, across different waves in the hyperbolic system is proposed here. A simple discretization for the non-conservative terms, consistent with the proposed Riemann solver, is employed. The efficacy of the solver is evaluated by solving a variety of two-phase test problems with large density ratios, viscous and surface tension effects. These evaluations are conducted on both structured and unstructured grids. The outcomes of the numerical experiments closely match with the analytical, experimental and numerical findings reported in the existing literature.