Integrated Sensing and Communications (ISAC) is emerging as a foundational paradigm for next-generation wireless networks, enabling communication infrastructures to simultaneously support data transmission and environment sensing. By tightly coupling radio sensing with communication functions, ISAC unlocks new capabilities for situational awareness, localization, tracking, and network adaptation. At the same time, the increasing scale, heterogeneity, and dynamics of future wireless systems demand self-organizing network intelligence capable of autonomously managing resources, topology, and services. Artificial intelligence (AI), particularly learning-driven and data-centric methods, has become a key enabler for realizing this vision. This survey provides a comprehensive and system-level review of AI-native ISAC-enabled self-organizing wireless networks. We develop a unified taxonomy that spans: (i) ISAC signal models and sensing modalities, (ii) network state abstraction and perception from sensing-aware radio data, (iii) learning-driven self-organization mechanisms for resource allocation, topology control, and mobility management, and (iv) cross-layer architectures integrating sensing, communication, and network intelligence. We further examine emerging learning paradigms, including deep reinforcement learning, graph-based learning, multi-agent coordination, and federated intelligence that enable autonomous adaptation under uncertainty, mobility, and partial observability. Practical considerations such as sensing-communication trade-offs, scalability, latency, reliability, and security are discussed alongside representative evaluation methodologies and performance metrics. Finally, we identify key open challenges and future research directions toward deployable, trustworthy, and scalable AI-native ISAC systems for 6G and beyond.