In the rapidly evolving landscape of medical documentation, transcribing clinical dialogues accurately is increasingly paramount. This study explores the potential of Large Language Models (LLMs) to enhance the accuracy of Automatic Speech Recognition (ASR) systems in medical transcription. Utilizing the PriMock57 dataset, which encompasses a diverse range of primary care consultations, we apply advanced LLMs to refine ASR-generated transcripts. Our research is multifaceted, focusing on improvements in general Word Error Rate (WER), Medical Concept WER (MC-WER) for the accurate transcription of essential medical terms, and speaker diarization accuracy. Additionally, we assess the role of LLM post-processing in improving semantic textual similarity, thereby preserving the contextual integrity of clinical dialogues. Through a series of experiments, we compare the efficacy of zero-shot and Chain-of-Thought (CoT) prompting techniques in enhancing diarization and correction accuracy. Our findings demonstrate that LLMs, particularly through CoT prompting, not only improve the diarization accuracy of existing ASR systems but also achieve state-of-the-art performance in this domain. This improvement extends to more accurately capturing medical concepts and enhancing the overall semantic coherence of the transcribed dialogues. These findings illustrate the dual role of LLMs in augmenting ASR outputs and independently excelling in transcription tasks, holding significant promise for transforming medical ASR systems and leading to more accurate and reliable patient records in healthcare settings.