MRI noninvasively quantifies liver fat and iron in terms of proton-density fat fraction (PDFF) and R2*. While conventional Cartesian-based methods require breath-holding, recent self-gated free-breathing radial techniques have shown accurate and repeatable PDFF and R2* mapping. However, data oversampling or computationally expensive reconstruction is required to reduce radial undersampling artifacts due to self-gating. This work developed an uncertainty-aware physics-driven deep learning network (UP-Net) that accurately and rapidly quantifies PDFF and R2* using data from self-gated free-breathing stack-of-radial MRI. UP-Net used an MRI physics loss term to guide quantitative mapping, and also provided uncertainty estimation for each quantitative parameter.