The investigation of neutrino interactions with matter serves as a valuable tool for understanding the fundamental structure of nucleons and potentially uncovering novel physics phenomena. To date, the neutrino-nucleon cross section has been examined across a range of energies spanning from a few hundred MeV to PeV. However, the pursuit of ultra-high-energy (UHE) cosmic neutrinos, surpassing 100 PeV in energy, holds the promise of further advancements. In the next 10-20 years, UHE neutrino telescopes, currently in the planning stage, may ultimately succeed in their detection. This article presents pioneering and comprehensive estimation forecasts for the ultra-high-energy neutrino-nucleon cross section, with a specific focus on the employment of neutrino radio-detection within the IceCube-Gen2 experiment. The study incorporates cutting-edge methodologies in UHE neutrino flux prediction, neutrino propagation within the Earth, radio detection techniques, and the treatment of background data to facilitate accurate cross section measurement projections. Assuming the successful detection of at least a few tens of UHE neutrino-induced events over a 10-year period, IceCube-Gen2 could achieve, for the first time, the measurement of the cross section at center-of-mass energies of approximately $\sqrt{s} \approx 10-100$ TeV. Furthermore, if the number of events exceeds one hundred, the precision of the cross section measurement could be comparable to its corresponding theoretical prediction.