Proton radiation therapy has the potential of achieving precise dose delivery to the tumor while sparing noncancerous surrounding tissue, owing to the sharp Bragg peaks of protons. Aligning the high dose region with the tumor requires accurate estimates of the proton stopping power ratio (SPR) of patient tissues, commonly derived from computed tomography (CT) image data. Photon-counting detectors within CT have demonstrated advantages over their energy-integrating counterparts, such as improved quantitative imaging, higher spatial resolution and filtering of electronic noise. In this study, the potential of photon-counting computed tomography for improving SPR estimation was assessed by training a deep neural network on a domain transform from photon-counting CT images to SPR maps. XCAT phantoms of the head were generated and used to simulate photon-counting CT images with CatSim, as well as to compute corresponding ground truth SPR maps. The CT images and SPR maps were then used as input and labels to a neural network. Prediction of SPR with the network yielded mean root mean square errors (RMSE) of 0.26-0.41 %, which is an improvement on errors reported for methods based on dual energy CT (DECT). These early results show promise for using a combination of photon-counting CT and deep learning for predicting SPR, which in extension demonstrates potential for reducing the beam range uncertainty in proton therapy.