Hospital-acquired infections (HAIs) pose a significant risk to global health, impacting millions of individuals globally. These infections have increased rates of morbidity and mortality due to the prevalence of widespread antimicrobial resistance (AMR). Graphene-based nanoparticles (GBNs) are known to possess extensive antimicrobial properties by inflicting damage to the cell membrane, suppressing virulence, and inhibiting microbial biofilms. Developing alternative therapies for HAIs and addressing AMR can be made easier and more affordable by combining nanoparticles with medicinal plants harboring antimicrobial properties. Hence, this study was undertaken to develop a novel graphene–silver nanocomposite via green synthesis using Trillium govanianum plant extract as a reducing agent. The resulting nanocomposite comprised silver nanoparticles embedded in graphene sheets. The antibacterial and antifungal properties of graphene–silver nanocomposites were investigated against several nosocomial pathogens, namely, Candida auris, Candida glabrata, Escherichia coli, Staphylococcus aureus, Klebsiella pneumoniae, and Pseudomonas aeruginosa. The nanocomposite displayed broad-range antimicrobial potential against the test pathogens, with minimum inhibitory concentrations (MICs) ranging between 31.25 and 125.0 µg/mL, and biofilm inhibition up to 80–96%. Moreover, nanocomposite-functionalized urinary catheters demonstrated hemocompatibility towards sheep erythrocytes and imparted anti-fouling activity to the biomaterial, while also displaying biocompatibility towards HEK 293 cells. Collectively, this investigation highlights the possible application of green-synthesized GBNs as an effective alternative to conventional antibiotics for combating multidrug-resistant pathogens.