We report the experimental realization of an optical trap that localizes single Cs atoms ≃215 nm from the surface of a dielectric nanofiber. By operating at magic wavelengths for pairs of counterpropagating red- and blue-detuned trapping beams, differential scalar light shifts are eliminated, and vector shifts are suppressed by ≈250. We thereby measure an absorption linewidth Γ/2π=5.7±0.1 MHz for the Cs 6S(1/2), F=4→6P(3/2), F'=5 transition, where Γ0/2π=5.2 MHz in free space. An optical depth d≃66 is observed, corresponding to an optical depth per atom d1≃0.08. These advances provide an important capability for the implementation of functional quantum optical networks and precision atomic spectroscopy near dielectric surfaces.