A general two-phase solver is developed to solve Navier-Stokes equation for low Mach number incompressible fluids on a mixed grid using a finite volume approach by employing a sharp fixed interface cut-cell method. This sharp interface divides the computational domain into two sub-domains corresponding to two different phases. The Navier-Stokes equations in each sub-domain are solved separately by coupling the boundary conditions for mass and momentum at the interface. These sub-domains can either be of same media (fluid-fluid) or different media (fluid-solid or fluid-gas). For a fluid-gas media, this enables to distinguish the sub-domains clearly and establish discontinuities in the thermophysical properties and a finite pressure jump based on the local curvature of the interface. Hence, this scheme retains accuracy for a wide range of density and viscosity ratios. In addition to algorithm development for cut-cell methodology, a novel time-advancement scheme is developed along the lines of Du Fort-Frankel explicit scheme for achieving conditional stability even with relaxed restrictions imposed by geometrical constraints. The novel time-advancement scheme is validated by comparing numerical and analytical solutions for a Taylor-Green vortex field and a detailed Von-Neumann stability analysis is performed for solving a linear advection-diffusion equation on a two-dimensional structured grid. The general two-phase solver for a fixed interface is then validated by first checking the order of accuracy of spatial operators in the Navier-Stokes for both linear and non-linear flows and then by performing validation for three distinct types of flows on a fixed grid, viz., single-phase flow on a mixed grid (lid-driven cavity flow); single-phase flow over an arbitrary solid body (flow over a cylinder in a driven cavity); and two-phase flow (static vapour bubble in liquid under equilibrium at zero gravity). In each of the aforementioned cases, unstructuredness in the computational domain arises due to user-specified cut-cells on the single-phase domain; body conformed grid at the interface of the arbitrary body and single phase domain, and interface distinguishing the two phases. The results are successfully validated with ones existing in the literature and a good agreement is observed.