Purpose: To propose a fully automated deep learning (DL) framework for the vertebral morphometry and Cobb angle measurement from three-dimensional (3D) computed tomography (CT) images of the spine, and validate the proposed framework on an external database. Methods: The vertebrae were first localized and segmented in each 3D CT image using a DL architecture based on an ensemble of U-Nets, and then automated vertebral morphometry in the form of vertebral body (VB) and intervertebral disk (IVD) heights, and spinal curvature measurements in the form of coronal and sagittal Cobb angles (thoracic kyphosis and lumbar lordosis) were performed using dedicated machine learning techniques. The framework was trained on 1725 vertebrae from 160 CT images and validated on an external database of 157 vertebrae from 15 CT images. Results: The resulting mean absolute errors (± standard deviation) between the obtained DL and corresponding manual measurements were 1.17 ± 0.40 mm for VB heights, 0.54 ± 0.21 mm for IVD heights, and 3.42 ± 1.36° for coronal and sagittal Cobb angles, with respective maximal absolute errors of 2.51 mm, 1.64 mm, and 5.52°. Linear regression revealed excellent agreement, with Pearson’s correlation coefficient of 0.943, 0.928, and 0.996, respectively. Conclusion: The obtained results are within the range of values, obtained by existing DL approaches without external validation. The results therefore confirm the scalability of the proposed DL framework from the perspective of application to external data, and time and computational resource consumption required for framework training.