Medical image registration is a critical process that aligns various patient scans, facilitating tasks like diagnosis, surgical planning, and tracking. Traditional optimization based methods are slow, prompting the use of Deep Learning (DL) techniques, such as VoxelMorph and Transformer-based strategies, for faster results. However, these DL methods often impose significant resource demands. In response to these challenges, we present NCA-Morph, an innovative approach that seamlessly blends DL with a bio-inspired communication and networking approach, enabled by Neural Cellular Automata (NCAs). NCA-Morph not only harnesses the power of DL for efficient image registration but also builds a network of local communications between cells and respective voxels over time, mimicking the interaction observed in living systems. In our extensive experiments, we subject NCA-Morph to evaluations across three distinct 3D registration tasks, encompassing Brain, Prostate and Hippocampus images from both healthy and diseased patients. The results showcase NCA-Morph's ability to achieve state-of-the-art performance. Notably, NCA-Morph distinguishes itself as a lightweight architecture with significantly fewer parameters; 60% and 99.7% less than VoxelMorph and TransMorph. This characteristic positions NCA-Morph as an ideal solution for resource-constrained medical applications, such as primary care settings and operating rooms.