In search settings, calibrating the scores during the ranking process to quantities such as click-through rates or relevance levels enhances a system's usefulness and trustworthiness for downstream users. While previous research has improved this notion of calibration for low complexity learning-to-rank models, the larger data demands and parameter count specific to modern neural text rankers produce unique obstacles that hamper the efficacy of methods intended for the learning-to-rank setting. This paper proposes exploiting large language models (LLMs) to provide relevance and uncertainty signals for these neural text rankers to produce scale-calibrated scores through Monte Carlo sampling of natural language explanations (NLEs). Our approach transforms the neural ranking task from ranking textual query-document pairs to ranking corresponding synthesized NLEs. Comprehensive experiments on two popular document ranking datasets show that the NLE-based calibration approach consistently outperforms past calibration methods and LLM-based methods for ranking, calibration, and query performance prediction tasks.