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We present a complete reevaluation of the irreducible two-loop vacuum-polarization correction to the photon propagator in quantum electrodynamics, i.e. with an electron-positron pair in the fermion propagators. The integration is carried out by reducing the integrations to a limited set of master integrals, which are calculated using integration-by-parts identities. Dimensional regularization is used in $D=4-2\epsilon$dimensions, and on-mass shell renormalization is employed. The one-loop effect is given to order $\epsilon$, to be combined with the $1/\epsilon$divergence of the two-loop amplitude. Master integrals are given. Final evaluations of two-loop energy shifts for $1S$, $2S$, and $2P$states are done analytically, and results are presented, with an emphasis on muonic hydrogen. For relativistic Dirac-Coulomb reference states, higher-order coefficients are obtained for the $Z\alpha$-expansion. We compare the results obtained to the existing literature. Published by the American Physical Society2024