Binary neutron star mergers (BNSMs) are currently the most promising source of r-process thanks to the detection of GW170817. The estimated occurring frequency and the amount of mass ejected per merger indicate that BNSMs by themselves can account for all the r-process enrichment in the Galaxy. However, the decreasing trend of [Eu/Fe] versus [Fe/H] of disk stars for [Fe/H] ≳ −1 in the solar neighborhood is inconsistent with the flat trend expected from BNSMs with a standard delay time distribution (DTD) ∝ t −1. This has led to the suggestion that either additional sources or modification to the DTD of BNSMs is required to match the observations. We investigate the effects of natal kicks received during the birth of neutron star binaries on the chemical evolution of r-process element Eu in the Milky Way by combining the results from the galactic dynamics code galpy with a one-zone Galactic chemical evolution model omega. We show that when key inputs from simulations of the inside-out disk evolution are combined with natal kicks, BNSMs can naturally reproduce the observed decreasing trend of [Eu/Fe] with [Fe/H] in the solar neighborhood without the need for modification to the DTD or additional r-process sources.