With satellite systems rapidly developing in multiple satellites, multiple tasks, and high-speed response speed requirements, existing computing techniques face the following challenges: insufficient computing power, limited computing resources, and weaker coordination ability. Meanwhile, most methods have more significant response speed and resource utilization limitations. To solve the above problem, we propose a distributed collaborative computing framework with a genetic algorithm-based task scheduling model (DCCF-GA), which can realize the collaborative computing between multiple satellites through genetic algorithm. Specifically, it contains two aspects of work. First, a distributed architecture of satellites is constructed where the main satellite is responsible for distribution and scheduling, and the computing satellite is accountable for completing the task. Then, we presented a genetic algorithm-based task scheduling model that enables multiple satellites to collaborate for completing the tasks. Experiments show that the proposed algorithm has apparent advantages in completion time and outperforms other algorithms in resource efficiency.