Recent years have witnessed a growing interest in Internet access from space assisted by low earth orbit (LEO) satellite networks. In the domain of the last-mile access for the Internet of Vehicles (IoV), hybrid free-space optical (FSO)/radio-frequency (RF) communication has recently attracted worldwide research efforts. While the transmission control protocol (TCP) is the most widely deployed transport protocol on the Internet, its performance in the error-prone environment of LEO satellite-assisted hybrid FSO/RF vehicular networks is not well understood. This paper develops a comprehensive analytical model based on the cross-layer approach for TCP performance, considering the FSO and RF satellite fading channels, modeled by the Gamma-Gamma and Nakagami-m distributions, respectively. The error-control solutions, including the Reed-Solomon (RS) code and Selective repeat automatic repeat request (SR-ARQ), are also employed. Numerical results quantitatively demonstrate the impact of transmission errors at lastmile links and different parameters/settings of error-control solutions on the TCP performance. The paper also supports the selection of proper TCP variants for the considered networks.INDEX TERMS Satellite networks, hybrid FSO/RF last-mile, Internet of vehicles, Transmission control protocols, Error-control solutions.