The interplay between incommensurability and strong correlations is a challenging open issue. It is explored here via numerical renormalization-group (NRG) study of models of a magnetic impurity in a one-dimensional quasicrystal. The principal goal is to elucidate the physics at the localization transition of the Aubry-André Hamiltonian, where a fractal spectrum and multifractal wave functions lead to a critical Aubry-André Anderson (AAA) impurity model with an energy-dependent multifractal hybridization function. This goal is reached in three stages of increasing complexity: (1) Anderson impurity models with uniform fractal hybridization functions are solved to arbitrarily low temperatures . Below a Kondo temperature, these models approach a fractal strong-coupling fixed point where impurity thermodynamic properties are oscillatory in about negative average values determined by the spectrum's fractal dimension , with set by the fractal self-similarity near the Fermi energy. (2) An impurity hybridizing uniformly with all conduction states of the critical AAA model is shown to approach the fractal strong-coupling fixed point corresponding to and . (3) When the multifractal wave functions of the critical AAA model are taken into account, low- impurity thermodynamic properties are again negative and oscillatory, but with a more complicated structure than in (2). Under sample-averaging, the mean and median Kondo temperatures exhibit power-law dependencies on the Kondo coupling with exponents characteristic of different fractal dimensions. We attribute these signatures to the impurity probing a distribution of fractal strong-coupling fixed points with decreasing temperature. To treat the AAA model, the numerical renormalization group (NRG) is combined with the kernel polynomial method (KPM) to form a general, efficient treatment of hosts without translational symmetry in arbitrary dimensions down to a temperature scale set by the KPM expansion order. Implications of our results for heavy-fermion quasicrystals and other applications of the KPM approach are discussed.