As one of the latest developments in X-ray computed tomography (CT), photon-counting technology allows spectral detection, demonstrating considerable advantages as compared to conventional CT. In this study, we investigated the use of a first-generation clinical photon-counting computed tomography (PCCT) scanner and estimated proton relative (to water) stopping power (RSP) of tissue-equivalent materials from virtual monoenergetic reconstructions provided by the scanner. A set of calibration and evaluation tissue-equivalent inserts were scanned at 120 kVp. Maps of relative electron density (RED) and effective atomic number (EAN) were estimated from the reconstructed virtual monoenergetic images (VMI) using an approach previously applied to a spectral CT scanner with dual-layer detector technology, which allows direct calculation of RSP using the Bethe-Bloch formula. The accuracy of RED, EAN, and RSP was evaluated by root-mean-square errors (RMSE) averaged over the phantom inserts. The reference RSP values were obtained experimentally using a water column in an ion beam. For RED and EAN, the reference values were calculated based on the mass density and the chemical composition of the inserts. Different combinations of low- and high-energy VMIs were investigated in this study, ranging from 40 to 190 keV. The overall lowest error was achieved using VMIs at 60 and 180 keV, with an RSP accuracy of 1.27% and 0.71% for the calibration and the evaluation phantom, respectively.