The pursuit of carbon-neutral wireless networks is increasingly constrained by the escalating energy demands of deep learning-based signal processing. Here, we introduce SpikACom (Spiking Adaptive Communications), a neuromorphic computing framework that synergizes brain-inspired spiking neural networks (SNNs) with wireless signal processing to deliver sustainable intelligence. SpikACom advances the paradigm shift from energy-intensive, continuous-valued processing to event-driven sparse computation. Moreover, it supports continual learning in dynamic wireless environments via a dual-scale mechanism that integrates channel distribution-aware context modulation with a synaptic consolidation rule using SNN-specific statistics, mitigating catastrophic forgetting. Evaluations across critical wireless communication tasks, including semantic communication, multiple-input multiple-output (MIMO) beamforming, and channel estimation demonstrate that SpikACom matches full-precision deep learning baselines while achieving an order-of-magnitude improvement in computational energy efficiency. Our results position SNNs as a promising pathway toward green wireless intelligence, providing evidence that neuromorphic computing can empower the sustainability of modern digital systems.