We present the results of molecular dynamics simulations of a nanoscale electrochemical cell. The simulations include an aqueous electrolyte solution with varying ionic strength (i.e., concentrations ranging from 0-4 M) between a pair of metallic electrodes held at constant potential difference. We analyze these simulations by computing the electrostatic potential profile of the electric double-layer region and find it to be nearly independent of ionic concentration, in stark contrast to the predictions of standard continuum-based theories. We attribute this lack of concentration dependence to the molecular influences of water molecules at the electrode-solution interface. These influences include the molecular manifestation of water's dielectric response, which tends to drown out the comparatively weak screening requirement of the ions. To support our analysis, we decompose water's interfacial response into three primary contributions: molecular layering, intrinsic (zero-field) orientational polarization, and the dipolar dielectric response.