Diet quality is a leading determinant of chronic disease risk. Advances in artificial intelligence (AI) have enabled food recommendation systems to adapt suggestions to user preferences and health goals. However, most current systems rely on loosely curated food databases and provide limited connection to a validated index. In this study, we propose a Healthy Eating Index (HEI) informed retrieval-augmented generation (RAG) framework that combines standardized nutrition databases with large language models (LLMs) for personalized food recommendations. Our proposed method anchors retrieval in the National Health and Nutrition Examination Survey (NHANES) and the Food Patterns Equivalents Database (FPED). A food-level embedding space is constructed from FPED-derived textual descriptions. For each entity, the system computes baseline HEI scores, retrieves candidate foods for intake recommendations, and estimates the HEI impact of simple substitutions or additions. A constrained RAG pipeline instantiated with a pretrained OpenAI LLM generates personalized recommendations and sources based on nutrient profiles and HEI contributions. The simulation results showed a mean HEI improvement of 6.45, with the proportion of users HEI over 50 increasing from 45.12 to 61.26. Quantile analysis revealed consistent improved shifts across the HEI distribution. Our findings suggest that the proposed LLM-RAG-based AI systems can support more precise, explainable, and personalized nutrition guidance to improve diet quality.