The Palatini gravitational action is enlarged by an arbitrary function f(X) of the determinants of the Ricci tensor and the metric, X = vertical bar det.R vertical bar/vertical bar det.g vertical bar The resulting Ricci-determinant theory exhibits novel deviations from general relativity. We study a particular realisation where the extension is characterized by the square-root of the Ricci determinant, f(X) = lambda(Edd) root X, which corresponds to the famous Eddington action. We analyze the obtained equations for perfect fluid source and show that the affine connection can be solved in terms of the energy density and pressure of the fluid through an obtained disformal metric. As an application, we derive the hydrostatic equilibrium equations for relativistic stars and inspect the significant effects induced by the square-root of the Ricci tensor. We find that an upper bound on lambda(Edd), at which deviations from the predictions of general relativity on neutron stars become prominent, corresponds to the hierarchy between the Planck and the vacuum mass scales. The Ricci-determinant gravity that we propose here is expected to have interesting implications in other cosmological domains.