Variable selection in high-dimensional nonlinear classification remains challenging due to the absence of explicit variable-wise structures. We propose Consistent Sparse Gradient Learning (CSGL), a nonparametric method that performs variable selection in a reproducing kernel Hilbert space by directly estimating the gradient of the Bayes decision function and imposing a functional group-lasso penalty on its components. We derive minimax-optimal convergence rates for the estimator and, under a restricted strong convexity condition, establish fast rates for general classification losses, overcoming the typical $n^{-1/2}$ barrier. We further prove selection consistency, using an operator-theoretic irrepresentable condition and adaptively weighted regularization to separate informative and noise variables. Computationally, we develop an efficient algorithm combining group-wise majorization descent with a strong sequential screening rule. Extensive simulations and real data analyses demonstrate that CSGL achieves superior prediction accuracy and stable variable recovery compared with existing linear and nonlinear competitors.