Passive acoustics provides a powerful tool for monitoring the endangered North Atlantic right whale ($Eubalaena$ $glacialis$), but robust detection algorithms are needed to handle diverse and variable acoustic conditions and differences in recording techniques and equipment. Here, we investigate the potential of deep neural networks for addressing this need. ResNet, an architecture commonly used for image recognition, is trained to recognize the time-frequency representation of the characteristic North Atlantic right whale upcall. The network is trained on several thousand examples recorded at various locations in the Gulf of St.\ Lawrence in 2018 and 2019, using different equipment and deployment techniques. Used as a detection algorithm on fifty 30-minute recordings from the years 2015-2017 containing over one thousand upcalls, the network achieves recalls up to 80%, while maintaining a precision of 90%. Importantly, the performance of the network improves as more variance is introduced into the training dataset, whereas the opposite trend is observed using a conventional linear discriminant analysis approach. Our work demonstrates that deep neural networks can be trained to identify North Atlantic right whale upcalls under diverse and variable conditions with a performance that compares favorably to that of existing algorithms.