By capturing complementary information from multiple omics data, multi-omics integration has demonstrated promising performance in disease prediction. However, with the increasing number of omics data, accurately quantifying the characterization ability of each omics and avoiding mutual interference becomes challenging due to the intricate relationships among omics data. Here, we propose a novel multi-omics integration framework that improves diagnostic prediction. Specifically, our framework involves constructing co-expression and co-methylation networks for each subject, followed by applying multi-level graph attention to incorporate biomolecule interaction information. The true-class-probability strategy is employed to evaluate omics-level confidence for classification, and the loss is designed using an adaptive mechanism to leverage both within- and across-omics information. Extensive experiments demonstrate that the proposed framework outperforms state-of-the-art methods on classification tasks, and indicate that the integration of three omics yields superior performance compared to employing only one or two data types. Code and data are available at https://github.com/Yaolab-fantastic/GRAMI-NET.