We consider the formation of ultrashort spatiotemporal optical waveforms in arrays of carbon nanotubes. We use a short-wave approximation to derive a generic two-dimensional sine-Gordon equation, describing ultrashort soliton evolution in such nanomaterials. This model was derived by using a rigorous application of the reductive perturbation formalism (multiscale analysis) for the Maxwell equations and for the corresponding Boltzmann kinetic equation for the distribution function of electrons in carbon nanotubes. We show numerically diffractionless and dispersionless robust propagation over large distances (with respect to the wavelength) of few-cycle (2+1)-dimensional spatiotemporal solitons in the form of optical breathers.