Low Earth Orbit (LEO) satellite networks are emerging as crucial components of space-air-ground integrated networks (SAGINs), extending beyond terrestrial capabilities to provide global data transmission services for the Internet of Things (IoT) and mobile devices. The proliferation of connected devices has led to increased data volumes and highly variable, bursty traffic patterns, thus posing significant challenges for network stability and necessitating effective routing control mechanisms. Traditional Lyapunov optimization methods have been fundamental in network optimization, offering stability guarantees under the assumption that traffic flows remain strictly within the network's capacity region. However, this assumption is often violated in LEO satellite networks due to their dynamic and bursty nature, thereby rendering conventional approaches inadequate for ensuring stability. To address this challenge, we propose a robust Lyapunov optimization framework tailored for LEO satellite networks. Our method relaxes the strict requirements of traditional Lyapunov optimization by allowing the network to tolerate finite violations of the capacity region while still ensuring overall system stability. This approach demonstrates that, for a stabilizable network system, it is not necessary for traffic to remain within the capacity region at every time slot. We validate the effectiveness of the proposed robust Lyapunov optimization through extensive simulations under various traffic conditions and LEO satellite network configurations. The results confirm that LEO satellite networks can maintain stability despite finite violations of the capacity region, ensuring reliable performance amid dynamic and bursty traffic demands.