In this paper, the growth stability of open-ended carbon nanotubes mediated by surface diffusion on the lateral surface of the nanotube is considered in detail. Nanotube growth and destabilization is viewed as a competition of two processes at the open growth edge: (i) hexagon formation sustaining the continuous growth of the regular hexagonal network, and (ii) thermally activated pentagon formation, which causes inward bending of the nanotube wall resulting in end closure, i.e., growth termination. The edge of the open-ended nanotube, if it is fed by a sufficiently large surface diffusion flux, may remain stable even without extrinsic stabilizing effects. The closure of the open end of the growing nanotube is shown to happen whenever a change in the growth conditions (temperature, carbon vapor pressure, or surface area from which the open end is fed) decreases the surface diffusion flux, and the characteristic time for new atom arrival on the edge becomes larger than the characteristic time for pentagon defect formation. These kinetic effects are also shown to define the transition from single wall to multiwall nanotube growth. Additionally, the effect of surface diffusion feeding nanotube growth from behind the growth interface is shown to stabilize open edge morphology, effectively smoothing the growth perturbations which may be caused by diffusion-limited aggregation at the edge.