To enhance the mechanical strength and cell adhesion of alginate hydrogel, making it satisfy the requirements of an ideal tissue engineering scaffold, the grafting of Arg-Gly-Asp (RGD) polypeptide sequence onto the alginate molecular chain was conducted by oxidation of sodium periodate and subsequent reduction amination of 2-methylpyridine borane complex (2-PBC) to synthesize alginate dialdehyde grafted RGD derivatives (ADA-RGD) with good cellular affinity. The interpenetrating network (IPN) composite hydrogels of alginate/polyvinyl alcohol/cellulose nanocrystals (ALG/PVA/CNCs) were fabricated through a physical mixture of ion cross-linking of sodium alginate (SA) with hydroxyapatite/D-glucono-δ-lactone (HAP/GDL), and physical cross-linking of polyvinyl alcohol (PVA) by a freezing/thawing method, using cellulose nanocrystals (CNCs) as the reinforcement agent. The effects of the addition of CNCs and different contents of PVA on the morphology, thermal stability, mechanical properties, swelling, biodegradability, and cell compatibility of the IPN composite hydrogels were investigated, and the effect of RGD grafting on the biological properties of the IPN composite hydrogels was also studied. The resultant IPN ALG/PVA/CNCs composite hydrogels exhibited good pore structure and regular 3D morphology, whose pore size and porosity could be regulated by adjusting PVA content and the addition of CNCs. By increasing the PVA content, the number of physical cross-linking points in PVA increased, resulting in greater stress support for the IPN composite hydrogels of ALG/PVA/CNCs and consequently improving their mechanical characteristics. The creation of the IPN ALG/PVA/CNCs composite hydrogels’ physical cross-linking network through intramolecular or intermolecular hydrogen bonding led to improved thermal resistance and reduced swelling and biodegradation rate. Conversely, the ADA-RGD/PVA/CNCs IPN composite hydrogels exhibited a quicker degradation rate, attributed to the elimination of ADA-RGD by alkali. The results of the in vitro cytocompatibility showed that ALG/0.5PVA/0.3%CNCs and ADA-RGD/PVA/0.3%CNCs composite hydrogels showed better proliferative activity in comparison with other composite hydrogels, while ALG/PVA/0.3%CNCs and ADA-RGD/PVA/0.3%CNCs composite hydrogels displayed obvious proliferation effects, indicating that PVA, CNCs, and ADA-RGD with good biocompatibility were conducive to cell proliferation and differentiation for the IPN composite hydrogels.