The method developed by the authors [Linear Algebra Appl. 277, No. 1-3, 199-236 (1998; Zbl 0932.44005)] for solving the truncated Hamburger matrix moment problem is extended for deriving a unified method for solving the matrix Stieltjes moment problem. The developed uniform method deals not only with both the nondegenerate and the degenerate cases but also with the cases when the given set of Hermitian matrices is either even or odd. As an application a complete solution of the Nevanlinna-Pick interpolation problem in the class of matrix-valued Stieltjes functions is also derived.