The present paper establishes the mean-field limit of the Abraham system for a system of identical charged particles subject to the electromagnetic interaction. This mean-field limit leads to a regularized variant of the relativistic Vlasov-Maxwell system. The result obtained here is similar to the one obtained in [F. Golse, Commun. Math. Phys. 310 (2012), 789-816], but is obtained by a purely Eulerian description of the particle dynamics.