The incorporation of silver nanoparticles (AgNPs) into alginate–gelatin (Alg-Gel) hydrogels can enhance the properties of these materials for bone regeneration applications, due to the antimicrobial properties of AgNPs and non-cytotoxic concentrations, osteoinductive properties, and regulation of stem cell proliferation and differentiation. Here, the hydrogel formulation included 2% (w/v) sodium alginate, 4 µg/mL AgNPs, and 2.5% (w/v) gelatin. AgNPs were synthesized using a 2% (w/v) aqueous extract of roasted green tea with silver nitrate. The aqueous extract of roasted green tea for AgNP synthesis was characterized using HPLC and UHPLC-ESI-QTOF-MS/MS, and antioxidant capacity was measured in Trolox equivalents (TE) from 4 to 20 nmol/well concentrations. Stem cells from human exfoliated deciduous tooth cells were used for differentiation assays including positive (SHEDs/hydrogel with AgNPs) and negative controls (hydrogel without AgNPs). FTIR was used for hydrogel chemical characterization. Statistical analysis (p < 0.05, ANOVA) confirmed significant findings. Roasted green tea extract contained caffeine (most abundant), (−)-Gallocatechin, gallic acid, and various catechins. XRD analysis revealed FCC structure, TEM showed quasispheroidal AgNPs (19.85 ± 3 nm), and UV–Vis indicated a plasmon surface of 418 nm. This integration of nanotechnology and biomaterials shows promise for addressing bone tissue loss in clinical and surgical settings.