One of the open issues of the standard cosmological model is the value of the cosmic dipole measured from the Cosmic Microwave Background (CMB), as well as from the number count of quasars and radio sources. These measurements are currently in tension, with the number count dipole being 2-5 times larger than expected from CMB measurements. This discrepancy has been pointed out as a possible indication that the cosmological principle is not valid. In this paper, we explore the possibility of detecting and estimating the cosmic dipole with gravitational waves (GWs) from compact binary mergers detected by the future next-generation detectors Einstein Telescope and Cosmic Explorer. We model the expected signal and show that for binary black holes, the dipole amplitude in the number count of detections is independent of the characteristics of the population and provides a systematic-free tool to estimate the observer velocity. We introduce techniques to detect the cosmic dipole from number counting of GW detections and estimate its significance. We show that a GW dipole consistent with the amplitude of the dipole in radio galaxies would be detectable with $>3\sigma$ significance with a few years of observation ($10^6$ GW detections) and estimated with a $16\%$ precision, while a GW dipole consistent with the CMB one would require at least $10^7$ GW events for a confident detection. We also demonstrate that a total number $N_{\rm tot}$ of GW detections would be able to detect a dipole with amplitude $v_o/c \simeq1/\sqrt{N_{\rm tot}}$.