Physical layer authentication (PLA) is the process of claiming identity of a node based on its physical layer characteristics such as channel fading or hardware imperfections. In this work, we propose a novel PLA method for the inter-satellite communication links (ISLs) of the LEO satellites. In the proposed PLA method, multiple receiving satellites validate the identity of the transmitter by comparing the Doppler frequency measurements with the reference mobility information of the legitimate transmitter and then fuse their decision considering the selected decision rule. Analytical expressions are obtained for the spoofing detection probability and false alarm probability of the fusion methods. Numerically obtained high authentication performance results pave the way to a novel and easily integrable authentication mechanism for the LEO satellite networks.