This paper presents a deep learning-aided iterative detection algorithm for massive overloaded multiple-input multiple-output (MIMO) systems where the number of transmit antennas $n$ is larger than that of receive antennas $m$. Since the proposed algorithm is based on the projected gradient descent method with trainable parameters, it is named the trainable projected gradient-detector (TPG-detector). The trainable internal parameters, such as the step-size parameter, can be optimized with standard deep learning techniques, i.e., the back propagation and stochastic gradient descent algorithms. This approach is referred to as data-driven tuning, and ensures fast convergence during parameter estimation in the proposed scheme. The TPG-detector mainly consists of matrix-vector product operations whose computational cost is proportional to $m n$ for each iteration. In addition, the number of trainable parameters in the TPG-detector is independent of the number of antennas. These features of the TPG-detector result in a fast and stable training process and reasonable scalability for large systems. Numerical simulations show that the proposed detector achieves a comparable detection performance to those of existing algorithms for massive overloaded MIMO channels, e.g., the state-of-the-art IW-SOAV detector, with a lower computation cost.