Contextual bandits are a central framework for sequential decision-making, with applications ranging from recommendation systems to clinical trials. While nonparametric methods can flexibly model complex reward structures, they suffer from the curse of dimensionality. We address this challenge using a single-index model, which projects high-dimensional covariates onto a one-dimensional subspace while preserving nonparametric flexibility. We first develop a nonasymptotic theory for offline single-index regression for each arm, combining maximum rank correlation for index estimation with local polynomial regression. Building on this foundation, we propose a single-index bandit algorithm and establish its convergence rate. We further derive a matching lower bound, showing that the algorithm achieves minimax-optimal regret independent of the ambient dimension $d$, thereby overcoming the curse of dimensionality. We also establish an impossibility result for adaptation: without additional assumptions, no policy can adapt to unknown smoothness levels. Under a standard self-similarity condition, however, we construct a policy that remains minimax-optimal while automatically adapting to the unknown smoothness. Finally, as the dimension $d$ increases, our algorithm continues to achieve minimax-optimal regret, revealing a phase transition that characterizes the fundamental limits of single-index bandit learning.