Data consisting of samples of probability density functions necessitate the development of methodologies that respect the inherent nonlinearities associated with densities. In many applications, density curves appear as functional response objects in a regression model with vector predictors. For such models, inference is key to understand the importance of density-predictor relationships, and the uncertainty associated with the estimated conditional mean densities, defined as conditional Fréchet means under a suitable metric. Using the Wasserstein geometry of optimal transport, we consider the Fréchet regression of density response curves and develop tests for covariate effects and simultaneous confidence bands for estimated conditional mean densities. The asymptotic behavior of these objects is based on underlying functional central limit theorems within Wasserstein space, and we demonstrate that they are asymptotically of the correct size and coverage, with uniformly strong consistency of the proposed tests under sequences of contiguous alternatives. These methods are illustrated through a regression analysis of post-intracerebral hemorrhage hematoma densities.