The European Spallation Source (ESS), currently under construction in Sweden, will produce an intense pulsed neutrino beam, facilitating high-statistics measurements of Coherent Elastic Neutrino-Nucleus Scattering (CE \(\nu \) NS) utilizing state-of-the-art detection technology. The remarkable precision and large event rates expected from the future ESS-CE \(\nu \) NS measurements will offer a unique opportunity to probe physics both within and beyond the Standard Model (SM) with enhanced sensitivity, surpassing current experimental constraints. In this study, we estimate the projected sensitivity of the expected CE \(\nu \) NS measurements at the ESS for determining the weak mixing angle and the neutron root mean square charge radius. Additionally, we investigate potential constraints on new physics scenarios, including neutrino non-standard interactions and the broader framework of generalized neutrino interactions.