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Depth completion aims to recover a dense depth map from the sparse depth data and the corresponding single RGB image. The observed pixels provide the significant guidance for the recovery of the unobserved pixels' depth. However, due to the sparsity of the depth data, the standard convolution operation, exploited by most of existing methods, is not effective to model the observed contexts with depth values. To address this issue, we propose to adopt the graph propagation to capture the observed spatial contexts. Specifically, we first construct multiple graphs at different scales from observed pixels. Since the graph structure varies from sample to sample, we then apply the attention mechanism on the propagation, which encourages the network to model the contextual information adaptively. Furthermore, considering the mutli-modality of input data, we exploit the graph propagation on the two modalities respectively to extract multi-modal representations. Finally, we introduce the symmetric gated fusion strategy to exploit the extracted multi-modal features effectively. The proposed strategy preserves the original information for one modality and also absorbs complementary information from the other through learning the adaptive gating weights. Our model, named Adaptive Context-Aware Multi-Modal Network (ACMNet), achieves the state-of-the-art performance on two benchmarks, {\it i.e.}, KITTI and NYU-v2, and at the same time has fewer parameters than latest models. Our code is available at: \url{https://github.com/sshan-zhao/ACMNet}.