Purpose: Proton radiation therapy may achieve precise dose delivery to the tumor while sparing non-cancerous surrounding tissue, owing to the distinct 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 for CT have demonstrated advantages over their energy-integrating counterparts, such as improved quantitative imaging, higher spatial resolution, and filtering of electronic noise. We assessed the potential of photon-counting computed tomography (PCCT) for improving SPR estimation by training a deep neural network on a domain transform from PCCT images to SPR maps. Approach: The XCAT phantom was used to simulate PCCT images of the head with CatSim, as well as to compute corresponding ground truth SPR maps. The tube current was set to 260 mA, tube voltage to 120 kV, and number of view angles to 4000. The CT images and SPR maps were used as input and labels for training a U-Net. Results: Prediction of SPR with the network yielded average root mean square errors (RMSE) of 0.26% to 0.41%, which was an improvement on the RMSE for methods based on physical modeling developed for single-energy CT at 0.40% to 1.30% and dual-energy CT at 0.41% to 3.00%, performed on the simulated PCCT data. Conclusions: These early results show promise for using a combination of PCCT and deep learning for estimating SPR, which in extension demonstrates potential for reducing the beam range uncertainty in proton therapy.