Deep learning applications at the network edge lead to a significant growth in AI-related carbon emissions, presenting a critical sustainability challenge. The existing edge computing frameworks optimize for latency and throughput, but they largely ignore the environmental impact of inference workloads. This paper introduces CarbonEdge, a carbon-aware deep learning inference framework that extends adaptive model partitioning with carbon footprint estimation and green scheduling apabilities. We propose a carbon-aware scheduling algorithm that extends traditional weighted scoring with a carbon efficiency metric, supporting a tunable performance--carbon trade-off (demonstrated via weight sweep). Experimental evaluations on Docker-simulated heterogeneous edge environments show that CarbonEdge-Green mode achieves a 22.9% reduction in carbon emissions compared to monolithic execution. The framework achieves 1.3x improvement in carbon efficiency (245.8 vs 189.5 inferences per gram CO2) with negligible scheduling overhead (0.03ms per task). These results highlight the framework's potential for sustainable edge AI deployment, providing researchers and practitioners a tool to quantify and minimize the environmental footprint of distributed deep learning inference.