This work aims to introduce and analyze an adaptive stabilized finite element method to solve a nonlinear Darcy equation with a pressure-dependent viscosity and mixed boundary conditions. We stated the discrete problem’s well-posedness and optimal error estimates, in natural norms, under standard assumptions. Next, we introduce and analyze a residual-based a posteriori error estimator for the stabilized scheme. Finally, we present some two- and three-dimensional numerical examples which confirm our theoretical results.