Spinal cord injury (SCI) is a devastating incidence leading to permanent paralysis and loss of sensory-motor functions potentially resulting in the formation of lesions within the spinal cord. Imaging biomarkers obtained from magnetic resonance imaging (MRI) scans can predict the functional recovery of individuals with SCI and help choose the optimal treatment strategy. Currently, most studies employ manual quantification of these MRI-derived biomarkers, which is a subjective and tedious task. In this work, we propose (i) a universal tool for the automatic segmentation of intramedullary SCI lesions, dubbed \texttt{SCIsegV2}, and (ii) a method to automatically compute the width of the tissue bridges from the segmented lesion. Tissue bridges represent the spared spinal tissue adjacent to the lesion, which is associated with functional recovery in SCI patients. The tool was trained and validated on a heterogeneous dataset from 7 sites comprising patients from different SCI phases (acute, sub-acute, and chronic) and etiologies (traumatic SCI, ischemic SCI, and degenerative cervical myelopathy). Tissue bridges quantified automatically did not significantly differ from those computed manually, suggesting that the proposed automatic tool can be used to derive relevant MRI biomarkers. \texttt{SCIsegV2} and the automatic tissue bridges computation are open-source and available in Spinal Cord Toolbox (v6.4 and above) via the \texttt{sct\_deepseg -task seg\_sc\_lesion\_t2w\_sci} and \texttt{sct\_analyze\_lesion} functions, respectively.