Recently, 4D Radar has emerged as a crucial sensor for 3D object detection in autonomous vehicles, offering both stable perception in adverse weather and high-density point clouds for object shape recognition. However, processing such high-density data demands substantial computational resources and energy consumption. We propose SpikingRTNH, the first spiking neural network (SNN) for 3D object detection using 4D Radar data. By replacing conventional ReLU activation functions with leaky integrate-and-fire (LIF) spiking neurons, SpikingRTNH achieves significant energy efficiency gains. Furthermore, inspired by human cognitive processes, we introduce biological top-down inference (BTI), which processes point clouds sequentially from higher to lower densities. This approach effectively utilizes points with lower noise and higher importance for detection. Experiments on K-Radar dataset demonstrate that SpikingRTNH with BTI significantly reduces energy consumption by 78% while achieving comparable detection performance to its ANN counterpart (51.1% AP 3D, 57.0% AP BEV). These results establish the viability of SNNs for energy-efficient 4D Radar-based object detection in autonomous driving systems. All codes are available at https://github.com/kaist-avelab/k-radar.