This note considers the problems of stability and stabilization for discretetime switched nonlinear systems with time-varying delay. The nonlinearity is assumed to satisfy a special constraint. The purpose of the robust stability problem is to give conditions such that the discrete-time switched nonlinear delay system is exponentially stable, while the purpose of stabilization is to design a state feedback control law such that the resulting closed-loop system is exponentially stable. By applying the average dwell time approach together with the piecewise Lyapunov function technique, also by constructing a proper Lyapunov-Krasovskii functional and employing the free-weighting matrix method, some delay-dependent stability conditions are proposed. A strict linear matrix inequality (LMI) design approach is developed. An explicit expression for the desired state feedback control law is also given. Finally, two numerical examples are provided to demonstrate the application of the proposed methods.