Background: A critical step in effective care and treatment planning for severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), the cause of the COVID-19 pandemic, is the assessment of the severity of disease progression. Chest x-rays (CXRs) are often used to assess SARS-CoV-2 severity, with two important assessment metrics being extent of lung involvement and degree of opacity. In this proof-of-concept study, we assess the feasibility of computer-aided scoring of CXRs of SARS-CoV-2 lung disease severity using a deep learning system. Materials and Methods: Data consisted of 396 CXRs from SARS-CoV-2 positive patient cases. Geographic extent and opacity extent were scored by two board-certified expert chest radiologists (with 20+ years of experience) and a 2nd-year radiology resident. The deep neural networks used in this study, which we name COVID-Net S, are based on a COVID-Net network architecture. 100 versions of the network were independently learned (50 to perform geographic extent scoring and 50 to perform opacity extent scoring) using random subsets of CXRs from the study, and we evaluated the networks using stratified Monte Carlo cross-validation experiments. Findings: The COVID-Net S deep neural networks yielded R$^2$ of 0.664 $\pm$ 0.032 and 0.635 $\pm$ 0.044 between predicted scores and radiologist scores for geographic extent and opacity extent, respectively, in stratified Monte Carlo cross-validation experiments. The best performing networks achieved R$^2$ of 0.739 and 0.741 between predicted scores and radiologist scores for geographic extent and opacity extent, respectively. Interpretation: The results are promising and suggest that the use of deep neural networks on CXRs could be an effective tool for computer-aided assessment of SARS-CoV-2 lung disease severity, although additional studies are needed before adoption for routine clinical use.