The relativistic local density approximation (LDA) for the Coulomb exchange functional in nuclear systems is presented. This approximation is composed of the well-known Slater approximation in the nonrelativistic scheme and the corrections due to the relativistic effects. Its validity in finite nuclei is examined by comparing with the exact treatment of the Coulomb exchange term in the relativistic Hartree-Fock-Bogoliubov theory. The relativistic effects are found to be important and the exact Coulomb exchange energies can be reproduced by the relativistic LDA within 5% demonstrated by the semimagic Ca, Ni, Zr, Sn, and Pb isotopes from proton drip line to neutron drip line.