When a biomaterial solution containing living cells is subject to bioprinting, the cells experience process-induced stresses, including shear and extensional stresses. These process-induced stresses breach cell membranes and can lead to cell damage, thus reducing cell viability and functioning within the printed constructs. Studies have been conducted to determine the influence of shear stress on cell damage; however, the effect of extensional stress has been typically ignored in the literature until the recently collected evidence of its importance. This paper presents a novel method to characterize and quantify the cell damage caused by both shear and extensional stresses in bioprinting. In this method, cell damage law is first established to relate cell damage to shear stress based on the experiments with a rheometer; the process-induced shear stress experienced by cells in bioprinting is represented, and the established cell damage model is applied to calculate the degree of cell damage caused by shear stress in bioprinting; then cell damage caused by extensional stress is inferred from the difference between the total cell damage and the amount of cell damage attributed to shear stress. With the obtained magnitude of extensional stress from fluidic simulation, the model that relates extensional stress to cell damage is established; the bioprinting process-induced cell damage attributed to both shear and extensional stresses is therefore presented. Schwann cells and myoblasts were used as examples to validate the models. Comparison between experimental and simulation results shows the effectiveness of the models presented in this paper. Moreover, the viability and proliferative ability of cells in the first 72 h after bioprinting is investigated, with the results illustrating that the process-induced forces affect not only cell viability but also their proliferative ability after bioprinting.