Bacterial infection is one of the principal reasons for the failure of tissue engineering scaffolds. Therefore, the development of multifunctional scaffolds that not only are able to guide tissue regeneration but also can inhibit bacterial colonization is of great importance for tissue engineering applications. In this study, a highly antibacterial, biocompatible, and biodegradable scaffold based on silk fibroin (SF) and gelatin methacryloyl (GelMA) was prepared. Sequential cross-linking of GelMA and SF under UV irradiation and methanol treatment, respectively, resulted in the formation of interpenetrating network (IPN) hydrogels with a porous structure. In addition, impregnation of the hydrogels with a nitric oxide (NO) donor molecule, S-nitroso-N-acetylpenicillamine (SNAP), led to the development of NO-releasing scaffolds with strong antibacterial properties. According to the obtained results, the addition of SF to GelMA hydrogels caused an enhancement in the mechanical properties and NO release kinetics and prevented their rapid enzymatic degradation in aqueous media. Furthermore, swelling the GelMA-SF scaffolds with SNAP resulted in a bacteria reduction efficiency of >99.9% against Gram-positive (Staphylococcus aureus) and Gram-negative (Escherichia coli) bacteria. The scaffolds also showed great cytocompatibility in vitro by increasing the proliferation and supporting the adhesion of 3T3 mouse fibroblast cells. Overall, GelMA-SF-SNAP showed great promise to be used as a scaffold for tissue engineering and wound healing applications.