One-dimensional (1-D) monotone systems have received considerable attention recently due to their wide applicability and interesting mathematical properties. One of these special properties is that, for LTI monotone systems, exponential stability is insensitive to time-delays. Some extensions to 1-D nonlinear monotone systems based on conditions of homogeneity have also been reported. In this paper, we study the problem of exponential stability of discrete-time 2-D nonlinear monotone systems described by the Roesser model with time-varying delays. Specifically, based on the monotone property and homogeneity of the associated vector fields, necessary and sufficient delay-independent exponential stability conditions are derived. The magnitudes of delays are also taken into deriving an explicit estimation of the exponential decay rate which correlates the impact of delays on the system performance. Two examples are given to demonstrate the effectiveness of the obtained results.