We investigate the effects of chirality on phonon thermal transport in semiconducting chiral single-walled carbon nanotubes (SWCNTs) using lattice dynamics combined with Boltzmann transport theory. We find that transverse acoustic and optical phonon modes, which are degenerate in nonchiral zigzag and armchair SWCNTs, are split in chiral SWCNTs, giving rise to finite phonon angular momentum associated with circular motion of individual atoms. This angular momentum is most efficiently generated in small-diameter nanotubes with intermediate chiral angles. Consequently, chiral SWCNTs are predicted to undergo thermally induced rigid-body rotation with an experimentally observable angular velocity via the thermal Einstein-de Haas effect.