Recent NMR experiments by Singer et al. [Singer, Phys. Rev. Lett. 95, 236403 (2005).] showed a deviation from Fermi-liquid behavior in carbon nanotubes with an energy gap evident at low temperatures. Here, a comprehensive theory for the magnetic field and temperature dependent NMR 13C spin-lattice relaxation is given in the framework of the Tomonaga-Luttinger liquid. The low temperature properties are governed by a gapped relaxation due to a spin gap ( approximately 30 K), which crosses over smoothly to the Luttinger liquid behavior with increasing temperature.