The sleep stage scoring allows the analysis and characterization of several sleep disorders. Since manual labeling is a tedious task subject to human errors, many proposals to perform this classification automatically have been made. Methods based on electroencephalogram (EEG) are the gold-standard, achieving the best results. However, they have complex instrumentation, which is a disadvantage for screening methods. For this reason, we propose an automatic sleep staging method using heart rate (HR) and peripheral oxygen saturation $(\mathbf{SpO}_{2})$ signals obtained from pulse oximeter, an ideal device for screening due to its low cost and simplicity. This method consists of two stacked layers of bidirectional gated recurrent units and a softmax layer to classify the output according to the American Academy of Sleep Medicine. To evaluate the performance, we use the Sleep Heart Health Study dataset, using 2500 HR and $\mathbf{SpO}_{2}$ signals corresponding to different patients for training, 1250 for validation, and 1250 for testing the models. The obtained results in the testing subset were 73.2 % for accuracy and 0.63 for the Cohen's Kappa coefficient. This performance shows that our model is able to outperform alternative methods that use cardiac signals from both pulse oximeter and electrocardiogram, but there is still an important gap to achieve the performances obtained using EEG.