Spoken question-answering (SQA) systems relying on automatic speech recognition (ASR) often struggle with accurately recognizing medical terminology. To this end, we propose MedSpeak, a novel knowledge graph-aided ASR error correction framework that refines noisy transcripts and improves downstream answer prediction by leveraging both semantic relationships and phonetic information encoded in a medical knowledge graph, together with the reasoning power of LLMs. Comprehensive experimental results on benchmarks demonstrate that MedSpeak significantly improves the accuracy of medical term recognition and overall medical SQA performance, establishing MedSpeak as a state-of-the-art solution for medical SQA. The code is available at https://github.com/RainieLLM/MedSpeak.