Charge-changing cross sections sigma(cc) of stable and unstable nuclei (Be9-11, C14-16, and O16-18) on a carbon target were investigated at 300 MeV/nucleon. A phenomenological analysis based on the Glauber theory indicates an approximate, but universal, scaling of sigma(cc) over a wide range of A/Z. This allows the determination of the density distributions of protons tightly bound in the nuclei. An application to C-16, which is considered to be an anomalously deformed nucleus, indicates a systematic evolution of proton root-mean-square radii and has revealed for the first time a neutron skin effect in carbon isotopes. Being complementary to isotope-shift and electron-scattering experiments, the present method can open up a new approach to explore the structure of exotic nuclei.