A hydrogel system based on oxidized alginate covalently crosslinked with gelatin (ADA-GEL) has been utilized for different biofabrication approaches to design constructs, in which cell growth, proliferation and migration have been observed. However, cell–bioink interactions are not completely understood and the potential effects of free aldehyde groups on the living cells have not been investigated. In this study, alginate, ADA and ADA-GEL were characterized via FTIR and NMR, and their effect on cell viability was investigated. In the tested cell lines, there was a concentration-dependent effect of oxidation degree on cell viability, with the strongest cytotoxicity observed after 72 h of culture. Subsequently, primary human cells, namely fibroblasts and endothelial cells (ECs) were grown in ADA and ADA-GEL hydrogels to investigate the molecular effects of oxidized material. In ADA, an extremely strong ROS generation resulting in a rapid depletion of cellular thiols was observed in ECs, leading to rapid necrotic cell death. In contrast, less pronounced cytotoxic effects of ADA were noted on human fibroblasts. Human fibroblasts had higher cellular thiol content than primary ECs and entered apoptosis under strong oxidative stress. The presence of gelatin in the hydrogel improved the primary cell survival, likely by reducing the oxidative stress via binding to the CHO groups. Consequently, ADA-GEL was better tolerated than ADA alone. Fibroblasts were able to survive the oxidative stress in ADA-GEL and re-entered the proliferative phase. To the best of our knowledge, this is the first report that shows in detail the relationship between oxidative stress-induced intracellular processes and alginate di-aldehyde-based bioinks.