Natural polysaccharides are highly attractive biopolymers recommended for medical applications due to their low cytotoxicity and hydrophilicity. Polysaccharides and their derivatives are also suitable for additive manufacturing, a process in which various customized geometries of 3D structures/scaffolds can be achieved. Polysaccharide-based hydrogel materials are widely used in 3D hydrogel printing of tissue substitutes. In this context, our goal was to obtain printable hydrogel nanocomposites by adding silica nanoparticles to a microbial polysaccharide’s polymer network. Several amounts of silica nanoparticles were added to the biopolymer, and their effects on the morpho-structural characteristics of the resulting nanocomposite hydrogel inks and subsequent 3D printed constructs were studied. FTIR, TGA, and microscopy analysis were used to investigate the resulting crosslinked structures. Assessment of the swelling characteristics and mechanical stability of the nanocomposite materials in a wet state was also conducted. The salecan-based hydrogels displayed excellent biocompatibility and could be employed for biomedical purposes, according to the results of the MTT, LDH, and Live/Dead tests. The innovative, crosslinked, nanocomposite materials are recommended for use in regenerative medicine.