Pure-phase cobalt-doped calcium hydroxyapatite ceramic samples with composition Ca10(PO4)6[(CoO2)x(OH)1-2x]2, where x = 0-0.2, were synthesized by high-temperature solid-state reaction, and their crystal structures, vibrational spectra, and magnetic properties were studied. Co atoms are found to enter into the apatite trigonal channel formally substituting H atoms and forming bent dioxocobaltate(II) anions. The anion exhibits single-molecule-magnet (SMM) behavior: slow relaxation of magnetization below 8 K under a nonzero magnetic field with an energy barrier of 63 cm-1. The barrier value does not depend on the concentration of Co ions, virtually coincides with the zero-field-splitting energy as determined from direct-current magnetization, and is very close to the value obtained earlier for cobalt-doped strontium hydroxyapatite. Moreover, the vibration frequencies of the dioxocobaltate(II) anion are found to be the same in calcium and strontium apatite matrixes. The very weak dependence of the SMM parameters on the matrix nature in combination with good chemical and thermal stabilities of the compounds provides wide opportunities to exploit the intrinsic properties of such a SMM-like anion.