The heavy quarkonia (charmonium c overlinec and bottomonium b overlineb) are investigated in a covariant formulation of the Salpeter equation. We parameterize confinement alternatively by a linearly rising scalar or a vector interaction kernel and take into account the one-gluon-exchange (OGE) interaction in the instantaneous approximation. Mass spectra as well as leptonic, two-photon, E1 and M1 decay widths are calculated. Our results show that a reasonable description of the experimental data can be obtained with both spin structures for the confining kernel. The relativistic treatment leads to an improved description compared to nonrelativistic results for the two-photon width of the ηc and to some extent for the E1-transition widths. However, characteristic deviations indicate that within a relativistic framework confinement is not described adequately by a simple potential in coordinate space.