This paper addresses the problem of the numerical computation of the stabilizing solution of a large class of stochastic game theoretic Riccati differential equations. A globally convergent iterative algorithm is proposed for this purpose. The main idea behind the proposed algorithm is to solve at each main iteration a system of uncoupled deterministic H-infinity-type Riccati equations. One of the main ingredients used in the proof of the convergence property is a new comparison theorem for this class of differential matrix equations. The performance of the proposed algorithm is illustrated through some numerical examples.