We discuss the internal structure of radially excited charmonium mesons based on the equal-time and Coulomb gauge Bethe-Salpeter (BS) amplitudes, which are obtained in lattice QCD. Our simulations are performed with a relativistic heavy-quark action for the charm quark on the (2+1)-flavor PACS-CS gauge configurations at the lightest pion mass, Mπ=156(7) MeV. The variational method is applied to the study of the optimal charmonium operators for ground and first excited states of S-wave charmonia. We successfully calculate the BS wave functions of ηc(2S) and ψ(2S) states, as well as ηc(1S) and J/ψ states, and then estimate the root-mean-square radii of both the 1S and 2S charmonium states. We also examine whether a series of the BS wave functions from the ground state to excited states can be described by a single set of the spin-independent and spin-dependent interquark potentials with a unique quark mass. It is found that the quark kinetic mass and both the central and the spin-spin charmonium potentials, determined from the 2S wave functions, fairly agree with the ones from the 1S wave functions. This strongly supports the validity of the potential description for the charmonium system - at least, below the open-charm threshold.