We theoretically investigate the current-induced excitation of torsional vibrations in a suspended carbon nanotube (CNT) quantum dot. By considering a CNT clamped between half-metallic ferromagnetic electrodes with an antiparallel magnetization configuration, we demonstrate that the spin-rotation coupling (SRC) enables the transfer of angular momentum from electron spins to the mechanical torsional mode under a constant source-drain voltage. Using a master-equation approach to analyze the coupled dynamics of the dot levels and a quantized torsional oscillator, we evaluate the steady-state current and phonon distribution. We find that when the Zeeman splitting matches the torsional phonon energy, the system exhibits a sharp resonant behavior in the current, accompanied by a significant increase in the phonon population. Our estimates for realistic device parameters indicate that this spin-driven mechanism can drive CNT torsional vibrations with detectable amplitudes. This work provides a theoretical basis for resonant current-controlled actuation of CNT torsional modes via SRC-mediated transfer of spin angular momentum.