The fully implicit approach is attractive in reservoir simulation for reasons of numerical stability and the avoidance of splitting errors when solving multiphase flow problems, but a large nonlinear system must be solved at each time step, so efficient and robust numerical methods are required to treat the nonlinearity. The Additive Schwarz Preconditioned Inexact Newton (ASPIN) framework, as an option for the outermost solver, successfully handles strong nonlinearities in computational fluid dynamics, but is barely explored for the highly nonlinear models of complex multiphase flow with capillarity, heterogeneity, and complex geometry. In this paper, the fully implicit ASPIN method is demonstrated for a finite volume discretization based on incompressible two-phase reservoir simulators in the presence of capillary forces and gravity. Numerical experiments show that the number of global nonlinear iterations is not only scalable with respect to the number of processors, but also significantly reduced compared with the standard inexact Newton method with a backtracking technique. Moreover, the ASPIN method, in contrast with the IMPES method, saves overall execution time because of the savings in timestep size.