The deposition and intercalation of metal atoms can induce superconductivity in monolayer and bilayer graphenes. For example, it has been experimentally proved that Li-deposited graphene is a superconductor with critical temperature Tc of 5.9 K, Ca-intercalated bilayer graphene C6CaC6 and K-intercalated epitaxial bilayer graphene C8KC8 are superconductors with Tc of 2-4 K and 3.6 K, respectively. However, the Tc of them are relatively low. To obtain higher Tc in graphene-based superconductors, here we predict a new Ca-intercalated bilayer graphene C2CaC2, which shows higher Ca concentration than the C6CaC6. It is proved to be thermodynamically and dynamically stable. The electronic structure, electron-phonon coupling (EPC) and superconductivity of C2CaC2 are investigated based on first-principles calculations. The EPC of C2CaC2 mainly comes from the coupling between the electrons of C-pz orbital and the high- and low-frequency vibration modes of C atoms. The calculated EPC constant λ of C2CaC2 is 0.75, and the superconducting Tc is 18.9 K, which is much higher than other metal-intercalated bilayer graphenes. By further applying -4% biaxial compressive strain to C2CaC2, the Tc can be boosted to 26.6 K. Thus, the predicted C2CaC2 provides a new platform for realizing superconductivity with the highest Tc in bilayer graphenes.