In recent years, the development of deep learning-based remaining useful life (RUL) prediction methods of bearings has flourished because of their high accuracy, easy implementation, and lack of reliance on a priori knowledge. However, there are two challenging issues concerning the prediction accuracy of existing methods. The run-to-failure sequential data and its RUL labels are almost inaccessible in real-world scenarios. Meanwhile, the existing models usually capture the general degradation trend of bearings while ignoring the local information, which restricts the model performance. To tackle the aforementioned problems, we propose a novel health indicator derived from the original vibration signals by combining principal components analysis with Euclidean distance metric, which was motivated by the desire to resolve the dependency on RUL labels. Then, we design a novel self-attention augmented convolution GRU network (SACGNet) to predict the RUL. Combining a self-attention mechanism with a convolution framework can both adaptively assign greater weights to more important information and focus on local information. Furthermore, Gated Recurrent Units are used to parse the long-term dependencies in weighted features such that SACGNet can utilize the important weighted features and focus on local features to improve the prognostic accuracy. The experimental results on the PHM 2012 Challenge dataset and the XJTU-SY bearing dataset have demonstrated that our proposed method is superior to the state of the art.