The properties of excitons in semiconducting single-walled carbon nanotubes (SWCNTs), isolated in vacuum or a medium, and their contributions to the optical spectra of nanotubes are studied within the elementary potential model, in which an exciton is represented as a bound state of two oppositely charged quasi-particles confined to the nanotube surface. The emphasis is given on the influence of dielectric environment surrounding a nanotube on the exciton spectra. For nanotubes in environment with permittivity less than 1.8 the ground-state exciton binding energies exceed the respective energy gaps, while the obtained binding energies of excitons in nanotubes in a medium with permittivity greater than 4 are in good accordance with the corresponding experimental data and consistent with known scaling relation for the environmental effect. The stabilization process of single-electron spectrum in SWCNTs in media with rather low permittivities is discussed.