This paper uses the classical theories of liquid crystal physics to formulate the Young-Laplace equation of capillary hydrostatics for interfaces between viscous isotropic (I) fluids and nematic liquid crystals (NLCs). The equation provides an expression of pressure jumps across interfaces and determines the geometry of the interface. In addition, the force balance equation at a capillary point or line singularity containing a nematic surface defect is also derived from first principles. It is shown that the areal pressure balance and the line or point force balance equations provide a complete set to determine the geometry of an interface containing a defect. A new expression for the Peach-Koehler force for surface defects is developed. The contribution of the surface Peach-Koehler force to areal pressure balances is also identified. The equations are applied to solve for the interfacial geometry of a NLC whose free-surface contains a point defect. Consistency of the new approach developed in this paper with previously presented results is demonstrated.