Towards performing a realistic autonomous minimally invasive spinal fixation procedure, in this paper, we introduce a unique robotic drilling system utilizing a concentric tube steerable drilling robot (CT-SDR) integrated with a seven degree-of-freedom robotic manipulator. The CT-SDR in integration with the robotic arm enables creating precise J-shape trajectories enabling access to the areas within the vertebral body that currently are not accessible utilizing existing rigid instruments. To ensure safety and accuracy of the autonomous drilling procedure, we also performed required calibration procedures. The performance of the proposed robotic system and the calibration steps were thoroughly evaluated by performing various drilling experiments on simulated Sawbone samples.