Calcium phosphate glasses (CPGs) are acquiring great importance in the biomedical field because of their thermomechanical and bioresorbable properties. In this study, optically transparent copper (1 mol %)-doped calcium phosphate glasses (CPGs_Cu) were prepared through the melt-quenching method, and their biocompatibility and antibacterial and antiviral properties were evaluated and compared with undoped CPGs. Biocompatibility was evaluated on murine fibroblast NIH-3T3 cells as a preliminary study of cytocompatibility. The in vitro tests were performed through indirect and direct cytotoxicity analyses by MTT and Alamar Blue assays and supported by electron microscopy observations. Microbiological analyses were performed against the most common Gram-negative and Gram-positive pathogens that cause nosocomial infections: Escherichia coli, Pseudomonas aeruginosa, Klebsiella pneumoniae, Staphylococcus aureus, and the methicillin-resistant Staphylococcus aureus strain. In addition, the bioglass samples were exposed to SARS-CoV-2 to assess their effects on viral survival. The obtained results assessed the biocompatibility of both bioglass types and their ability to reduce the viral load and trap the virus. In addition, Cu2+-doped bioglass was found to be antibacterial despite its low content (1 mol %) of copper, making this a promising candidate material for biomedical applications, e.g., surgery probes, drug delivery, and photodynamic therapy.