Progress in the synthesis of semiconductor nanowires (NWs) has prompted intensive inquiry into understanding the science of their growth mechanisms and ultimately the technological applications they promise. We present new results for the size-dependent growth kinetics of Ge NWs and correlate the results with a direct experimental measurement of the Gibbs-Thomson effect, a measured increase in the Ge solute concentration in liquid Au-Ge droplets with decreasing diameter. This nanoscale-dependent effect emerges in vapor-liquid-solid Ge NW growth and leads to a decrease in the NW growth rate for smaller diameter NWs under a wide range of growth conditions with a cutoff in growth at sufficiently small sizes. These effects are described quantitatively by an analytical model based on the Gibbs-Thomson effect. A comprehensive treatment is provided and shown to be consistent with experiment for the effect of NW growth time, temperature, pressure, and doping on the supersaturation of Ge in Au, which determines the growth rate and critical cutoff diameter for NW growth. These results support the universal applicability of the Gibbs-Thomson effect to sub-100 nm diameter semiconductor NW growth.