There has been significant progress in implementing deep learning models in disease diagnosis using chest X- rays. Despite these advancements, inherent biases in these models can lead to disparities in prediction accuracy across protected groups. In this study, we propose a framework to achieve accurate diagnostic outcomes and ensure fairness across intersectional groups in high-dimensional chest X- ray multi-label classification. Transcending traditional protected attributes, we consider complex interactions within social determinants, enabling a more granular benchmark and evaluation of fairness. We present a simple and robust method that involves retraining the last classification layer of pre-trained models using a balanced dataset across groups. Additionally, we account for fairness constraints and integrate class-balanced fine-tuning for multi-label settings. The evaluation of our method on the MIMIC-CXR dataset demonstrates that our framework achieves an optimal tradeoff between accuracy and fairness compared to baseline methods.