Graph neural networks have shown impressive capabilities in solving various graph learning tasks, particularly excelling in node classification. However, their effectiveness can be hindered by the challenges arising from the widespread existence of noisy measurements associated with the topological or nodal information present in real-world graphs. These inaccuracies in observations can corrupt the crucial patterns within the graph data, ultimately resulting in undesirable performance in practical applications. To address these issues, this paper proposes a novel uncertainty-aware graph learning framework motivated by distributionally robust optimization. The framework aims to alleviate the challenges by considering the distributional uncertainty associated with the graph data. Specifically, we use a graph neural network-based encoder to embed the node features and find the optimal node embeddings by minimizing the worst-case risk through a minimax formulation. Such an uncertainty-aware learning process leads to improved node representations and a more robust graph predictive model that effectively mitigates the impact of uncertainty arising from data noise. The learned LFDs also provide a means to quantify the predictive uncertainty, which is valuable in some uncertainty-sensitive scenarios where incorrect decisions can have severe consequence. In addition, we adopt the idea of differentiable optimization and develop an end-to-end learning algorithm that seamlessly integrates graph learning and distributionally robust optimization. Our experimental result shows that the proposed framework achieves superior predictive performance compared to the state-of-the-art baselines under various noisy settings.