Multi-view multi-label classification has recently received extensive attention due to its wide-ranging applications across various fields, such as medical imaging and bioinformatics. However, views and labels are usually incomplete in practical scenarios, attributed to the uncertainties in data collection and manual labeling. To cope with this issue, we propose an uncertainty-aware pseudo-labeling and dual graph driven network (UPDGD-Net), which can fully leverage the supervised information of the available labels and feature information of available views. Different from the existing works, we leverage the label matrix to impose dual graph constraints on the embedded features of both view-level and label-level, which enables the method to maintain the inherent structure of the real data during the feature extraction stage. Furthermore, our network incorporates an uncertainty-aware pseudo-labeling strategy to fill the missing labels, which not only addresses the learning issue of incomplete multi-labels but also enables the method to explore more reliable supervised information to guide the network training. Extensive experiments on five datasets demonstrate that our method outperforms other state-of-the-art methods.