As the representations output by Graph Neural Networks (GNNs) are increasingly employed in real-world applications, it becomes important to ensure that these representations are fair and stable. In this work, we establish a key connection between fairness and stability and leverage it to propose a novel framework, NIFTY (uNIfying Fairness and stabiliTY), which can be used with any GNN to learn fair and stable representations. We introduce an objective function that simultaneously accounts for fairness and stability and proposes layer-wise weight normalization of GNNs using the Lipschitz constant. Further, we theoretically show that our layer-wise weight normalization promotes fairness and stability in the resulting representations. We introduce three new graph datasets comprising of high-stakes decisions in criminal justice and financial lending domains. Extensive experimentation with the above datasets demonstrates the efficacy of our framework.