Existing urban navigation algorithms employ integrity monitoring (IM) to mitigate the impact of measurement bias errors and determine system availability when estimating the position of a receiver. Many IM techniques, such as receiver autonomous integrity monitoring (RAIM), utilize measurement residuals associated with a single receiver position to provide integrity. However, identifying a single correct receiver position is often challenging in urban environments due to low satellite visibility and multiple measurements with bias errors. To address this, we propose Particle RAIM as a novel framework for robust state estimation and IM using GNSS and odometry measurements. Particle RAIM integrates residual-based RAIM with a particle filter and Gaussian mixture model likelihood to jointly perform state estimation and fault mitigation using a multimodal probability distribution of the receiver state. Our experiments on simulated and real-world data show that Particle RAIM achieves smaller positioning errors as well as smaller probability of false alarm and probability of missed-identification in determining system availability than existing urban localization and IM approaches in challenging environments with a relatively small computation overhead.