<p>We develop and analyze an inexact regularized alternating projection method for nonconvex feasibility problems. Such a method employs inexact projections on one of the two sets, according to a set of well-defined conditions. We prove the global convergence of the algorithm, provided that a certain merit function satisfies the Kurdyka-Łojasiewicz property on its domain. The method is then specialized to the class of affine rank minimization problems, which includes matrix completion as a special case. We approximate the truncated Singular Value Decomposition of the matrix that has to be projected by means of a Krylov solver, and provide suitable stopping criteria for the Krylov method complying with the theoretical inexactness conditions. The information needed to implement such stopping criteria do not require an extra computational effort as they are a by-product of the Krylov method itself and avoid the so called oversolving phenomena. Results of the numerical validation of the algorithm on matrix completion problems are presented.</p>