Two-stage distributionally robust stochastic noncooperative games with continuous decision variables are studied. In such games, each player solves a two-stage distributionally robust optimization problem depending on the decisions of the other players. Existing studies in this area have been limited with strict assumptions, such as linear decision rules, and supposed that each player solves a two-stage linear distributionally robust optimization with a specifically structured ambiguity set. This limitation motivated us to generalize and analyze the game in a nonlinear case. The contributions of this study are (i) demonstrating the conditions for the existence of two-stage Nash equilibria under convexity and compactness assumptions, and (ii) consideration of a two-stage distributionally robust Cournot--Nash competition as an application, as well as an investigation into the conditions for the existence of market equilibria in an economic sense. We also report some results of numerical experiments to illustrate how distributional robustness affects the decision of each player in the Cournot--Nash competition.