Satellite-based microservice systems have emerged as a promising architecture for enabling scalable and flexible service deployment in Low Earth Orbit (LEO) satellite networks, which are increasingly recognized as a key solution to meet the rising demand for global communication and computation, especially in remote and underserved regions. However, routing microservices efficiently in such systems presents major challenges due to the dynamic topology, intermittent connectivity, and unstable link conditions inherent to satellite constellations. These issues become even more severe under high traffic loads. To address these challenges, we propose Service Pressure, a novel routing algorithm specifically designed for satellite-based microservice systems. Service Pressure comprises two key components: first, the construction of an augmented subgraph to model the complex execution and data transmission dependencies of microservices; second, a distributed service routing algorithm that utilizes queue backlogs. This combination enables the algorithm to effectively handle high throughput and adapt to the dynamic network conditions of satellite constellations. By optimizing resource utilization, minimizing latency, and balancing load across satellite nodes, Service Pressure ensures efficient and stable service orchestration even under fluctuating traffic conditions. Extensive simulations demonstrate that our approach significantly outperforms existing routing methods, particularly in terms of throughput, latency, and stability. Service Pressure offers a significant advancement in satellite microservice routing, making it ideal for next-generation space-ground integrated networks.