Deep graph learning models have demonstrated remarkable capabilities in processing graph-structured data and have been widely applied across various fields. However, their complex internal architectures and lack of transparency make it difficult to explain their decisions, resulting in opaque models that users find hard to understand and trust. In this paper, we explore model-level explanation techniques for deep graph learning models, aiming to provide users with a comprehensive understanding of the models' overall decision-making processes and underlying mechanisms. Specifically, we address the problem of counterfactual explanations for deep graph learning models by introducing a generative model-level counterfactual explanation approach called GCFX, which is based on deep graph generation. This approach generates a set of high-quality counterfactual explanations that reflect the model's global predictive behavior by leveraging an enhanced deep graph generation framework and a global summarization algorithm. GCFX features an architecture that combines dual encoders, structure-aware taggers, and Message Passing Neural Network decoders, enabling it to accurately learn the true latent distribution of input data and generate high-quality, closely related counterfactual examples. Subsequently, a global counterfactual summarization algorithm selects the most representative and comprehensive explanations from numerous candidate counterfactuals, providing broad insights into the model's global predictive patterns. Experiments on a synthetic dataset and several real-world datasets demonstrate that GCFX outperforms existing methods in terms of counterfactual validity and coverage while maintaining low explanation costs, thereby offering crucial support for enhancing the practicality and trustworthiness of global counterfactual explanations.