The goal of digital contact tracing is to diminish the spread of an epidemic or pandemic by detecting and mitigating public health emergencies using digital technologies. Since the start of the COVID-$19$ pandemic, a wide variety of mobile digital apps have been deployed to identify people exposed to the SARS-CoV-2 coronavirus and to stop onward transmission. Tracing sources of spreading (i.e., backward contact tracing), has been used in Japan and Australia, has proven crucial as going backward can pick up infections that might otherwise be missed at superspreading events. Can network science robustify backward contact tracing? How to automatically optimize data-driven contact tracing? This will entail a learning by design approach. In this paper, we formulate the forward and backward contact tracing problem for contagion source inference as a maximum-likelihood (ML) estimation problem with iterative infection networked data sampling. This covers a well-known special case of a priori knowing the entire infection network topology. As the full infection network topology is rarely accurately known upfront, this problem is challenging to solve due to its combinatorial complexity, problem scale and the speed of spreading as a disease virus becomes more infectious over time (e.g., the COVID-19 Omicron variant). We propose a Graph Neural Network (GNN) learning framework, DeepTrace, to efficiently compute the ML estimator and to track the most-likely superspreader iteratively as the contact tracing networked data grows in size. The GNN supervised learning leverages the likelihood-estimation structure so as to endow the training set with useful topological features to facilitate accelerated training time and convergence behavior. Using data of COVID-19 variants with different infectiousness property, we show that DeepTrace has better performance than the state-of-the-art for identifying superspreaders and can serve as a statistically rigorous basis to design scalable digital contact tracing for future epidemics.