Logic synthesis is a crucial phase in the circuit design process, responsible for transforming hardware description language (HDL) designs into optimized netlists. However, traditional logic synthesis methods are computationally intensive, restricting their iterative use in refining chip designs. Recent advancements in large language models (LLMs), particularly those fine-tuned on programming languages, present a promising alternative. In this paper, we introduce VeriDistill, the first end-to-end machine learning model that directly processes raw Verilog code to predict circuit quality-of-result metrics. Our model employs a novel knowledge distillation method, transferring low-level circuit insights via graphs into the predictor based on LLM. Experiments show VeriDistill outperforms state-of-the-art baselines on large-scale Verilog datasets and demonstrates robust performance when evaluated on out-of-distribution datasets.