  We report Resonant Inelastic X-ray Scattering (RIXS) study of the magnetic
excitation spectrum in a highly insulating Eu$_{2}$Ir$_{2}$O$_{7}$ single
crystal that exhibits a metal-insulator transition at $T_{MI}$ = 111(7) K. A
propagating magnon mode with 20 meV bandwidth and 28 meV magnon gap is found in
the excitation spectrum at 7 K, which is expected in the all-in-all-out (AIAO)
magnetically ordered state. This magnetic excitation exhibits substantial
softening as temperature is raised towards $T_{MI}$, and turns into highly
damped excitation in the paramagnetic phase. Remarkably, the softening occurs
throughout the whole Brillouin zone including the zone boundary. This
observation is inconsistent with magnon renormalization expected in a local
moment system, and indicates that the strength of electron correlation in
Eu$_{2}$Ir$_{2}$O$_{7}$ is only moderate, so that electron itinerancy should be
taken into account in describing its magnetism.
