Injectable hydrogels, as carriers, offer great potential to incorporate cells or growth factors for dental tissue regeneration. Notably, the development of injectable hydrogels with appropriate structures and properties has been a challenging task, leaving much to be desired in terms of cytocompatibility, antibacterial and self-healing properties, as well as the ability to support dental stem cell functions. This paper presents our study on the development of a novel self-cross-linkable hydrogel composed of oxidized alginate and carboxymethyl chitosan and its characterization as a cell carrier for dental enamel regeneration in vitro. Oxidized alginate was synthesized with 60% theoretical oxidation degree using periodate oxidation and characterized by Fourier Transform Infrared spectroscopy, proton nuclear magnetic resonance spectroscopy, and Ultraviolet-visible absorption spectroscopy. Then, hydrogels were prepared at three varying weight ratios of oxidized alginate to carboxymethyl chitosan (4:1, 3:1, and 2:1) through Schiff base reactions, which was confirmed by Fourier Transform Infrared spectroscopy. The hydrogels were characterized in terms of gelation time, swelling ratio, structure, injectability, self-healing, antibacterial properties, and in vitro characterization for enamel regeneration. The results demonstrated that, among the three hydrogels examined, the one with the highest ratio of oxidized alginate (i.e., 4:1) had the fastest gelation time and the lowest swelling ability, and that all hydrogels were formed with highly porous structures and were able to be injected through a 20-gauge needle without clogging. The injected hydrogels could be rapidly reformed with the self-healing property. The hydrogels also showed antibacterial properties against two cariogenic bacteria: Streptococcus mutans and Streptococcus sobrinus. For in vitro enamel regeneration, a dental epithelial cell line, HAT-7, was examined, demonstrating a high cell viability in the hydrogels during injection. Furthermore, HAT-7 cells encapsulated in the hydrogels showed alkaline phosphatase production and mineral deposition, as well as maintaining their round morphology, after 14 days of in vitro culture. Taken together, this study has provided evidence that the oxidized alginate-carboxymethyl chitosan hydrogels could be used as an injectable cell carrier for dental enamel tissue engineering applications.