The current-induced domain-wall depinning in ferromagnetic nanowires with perpendicular magnetic anisotropy is investigated. The nanowire patterns with 500-nm width are fabricated from 8-nm-thick Co-Re-Pt alloy films on 1-nm-thick Ta underlayer. By injecting the current into the nanowire, the domain wall motion via depinning from extrinsic structural pinning sites is observed. The depinning time ranging over 1 mus-1 s is found to be exponentially dependent on the current density. Based on generalized Neel-Brown formula, the slope of the exponential dependence is analyzed to estimate the depinning energy barrier as a function of the current density under a constant bias magnetic field.