Deep unfolding networks are rapidly gaining attention for solving imaging inverse problems. However, the computational and memory complexity of existing deep unfolding networks scales with the size of the full measurement set, limiting their applicability to certain large-scale imaging inverse problems. We propose SCRED-Net as a novel methodology that introduces a stochastic approximation to the unfolded regularization by denoising (RED) algorithm. Our method uses only a subset of measurements within each cascade block, making it scalable to a large number of measurements for efficient end-to-end training. We present numerical results showing the effectiveness of SCRED-Net on intensity diffraction tomography (IDT) and sparse-view computed tomography (CT). Our results show that SCRED-Net matches the performance of a batch deep unfolding network at a fraction of training and operational complexity.