We present a fully relativistic extension of the optimized-potential method (ROPM), including the transverse electron-electron interaction and vacuum corrections. Using perturbation theory on the basis of the Kohn-Sham Hamiltonian an exact representation of the relativistic exchange-correlation energy ${E}_{\mathrm{xc}}$ in terms of Kohn-Sham orbitals and eigenvalues is derived. The most simple, viable approximation to this ${E}_{\mathrm{xc}}$ is obtained by a second-order expansion in powers of ${e}^{2},$ which leads to a M\o{}ller-Plesset-type correlation functional ${E}_{\mathrm{c}}^{(2)}.$ Due to this origin ${E}_{\mathrm{c}}^{(2)}$ allows a first-principles, seamless description of long-range dispersive forces. The ROPM integral equation that determines the full exchange-correlation four potential ${v}_{\mathrm{xc}}^{\ensuremath{\mu}}$ is presented, and specified in detail for ${E}_{\mathrm{c}}^{(2)}.$ We also analyze the Krieger-Li-Iafrate (KLI) approximation to the exact ROPM integral equation, pointing out an inherent ambiguity of the KLI approximation which arises for eigenvalue-dependent ${E}_{\mathrm{xc}}.$ The gauge properties of ${E}_{\mathrm{xc}}$ and the ROPM integral equation are discussed by examining the transversality of the Kohn-Sham current-current response function. It is demonstrated that due to the multiplicative nature of the total effective potential the density functional definition of the no-pair transverse exchange energy guarantees gauge invariance, in contrast to the relativistic Hartree-Fock scheme. On the other hand, the correlation energy is gauge dependent as soon as the no-pair approximation is applied. In addition, we show that the no-pair approximation automatically implies a definite intrinsic gauge for the spatial components of ${v}_{\mathrm{xc}}^{\ensuremath{\mu}}.$ The significance of the self-consistent treatment of the transverse interaction for heavy atoms is investigated numerically within the exchange-only limit. By comparing self-consistent with first-order perturbative inclusion of the transverse exchange it is shown that second-order transverse corrections cannot be neglected in calculations of ground state or inner-shell transition energies of heavy atoms, if one aims at spectroscopic accuracy. It is furthermore found that the Breit approximation for the full transverse interaction is not as accurate for the exchange potential as it is for the exchange energy. Finally, the KLI approximation is examined numerically, thereby resolving the ambiguity for the case of the transverse exchange.