Facilitated by the emergence of neuromorphic hardware, neuromorphic algorithms mimic the brain’s asynchronous computation to improve energy efficiency, low latency, and robustness, which are crucial for a wide variety of real-time robotic applications. However, the limited on-chip learning abilities hinder the applicability of neuromorphic computing to real-world robotic tasks. Biomimetism can overcome this limitation by complementing or replacing training with the knowledge of the brain’s connectome associated with the targeted behavior. By drawing inspiration from the human oculomotor network, we designed a spiking neural network (SNN) that tracked visual targets in real-time. We deployed the biomimetic controller on Intel’s Loihi neuromorphic processor to control an in-house robotic head. The robot’s behavior resembled the smooth pursuit and saccadic eye movements observed in humans, while the SNN on Loihi exhibited similar performance to a CPU-run PID controller. Interestingly, this behavior emerged from the SNN without training, which places the biomimetic design as an alternative to the energy- and data-greedy learning-based methods. This work reinforces our on-going efforts to devise energy-efficient autonomous robots that mimic the robustness and versatility of their biological counterparts.