The imminent integration of autonomous vehicles and mobile robots in urban settings presents a critical safety challenge for future intelligent transportation systems. This paper addresses the complex problem of coordinating heterogeneous agents with disparate dynamics at unregulated intersections. We introduce a novel framework, differentiable model predictive safety (DMPS), which embeds the foresight of model-predictive control into a data-driven, end-to-end reinforcement learning architecture. DMPS agents learn a latent dynamics model to predict future trajectories contingent on their actions. A learned, differentiable safety critic then evaluates the risk of these trajectories. Crucially, by leveraging backpropagation through the entire unrolled predictive model, agents can efficiently compute the gradient of future safety with respect to their current action, enabling a minimal and precise online safety correction. Integrated into a multi-agent training scheme, DMPS virtually eliminates collisions to less than 5.6% in high-density, mixed vehicle-robot traffic simulations, demonstrating state-of-the-art safety without compromising energy and traffic efficiency.