Autonomous obstacle avoidance is a critical capability for underwater robots to operate safely and sustainably in complex, unfamiliar, and unknown underwater environments. Existing methods often lack information processing and intelligent rapid decision-making ability similar to the human brain, making it difficult to adapt to the complex and challenging underwater environment. To address these limitations, with the spherical underwater robot (SUR) as the research object, a novel brain-inspired spiking neural network, neuromorphic hybrid deep deterministic policy gradient (Neuro-HDDPG), is proposed in this paper. The soft reset membrane potential update mechanism is designed to better represent the variation of spiking neuron membrane potentials. By integrating the spiking neural network and deep reinforcement learning, the proposed Neuro-HDDPG is composed of a soft reset spiking actor normal network (SANN) and deep critic normal network (DCNN). The SANN consists of soft reset improved leaky integrate-and-fire (SR-ILIF) neurons, and the DCNN comprises artificial neurons, realizing autonomous obstacle avoidance exploration of SUR in complex and unknown environments, with more temporal continuity and biological interpretability. To evaluate the obstacle avoidance efficiency of the proposed Neuro-HDDPG, through the ablation studies and comparison experiments with other known methods, the proposed Neuro-HDDPG achieved the highest success rate of 91% and 92%, respectively, in the two underwater evaluation environments with different levels of complexity, demonstrating superior obstacle avoidance performance and forming a reliable and efficient underwater obstacle avoidance decision-making capability. Simultaneously, the concept of combining spiking neural network with deep reinforcement learning provides an intelligent and reliable reference for other unmanned underwater intelligent systems.