Studying spin-momentum correlations in hadronic collisions offers a glimpse into a three-dimensional picture of proton structure. The transverse single-spin asymmetry for midrapidity isolated direct photons in p^{↑}+p collisions at sqrt[s]=200 GeV is measured with the PHENIX detector at the Relativistic Heavy Ion Collider (RHIC). Because direct photons in particular are produced from the hard scattering and do not interact via the strong force, this measurement is a clean probe of initial-state spin-momentum correlations inside the proton and is in particular sensitive to gluon interference effects within the proton. This is the first time direct photons have been used as a probe of spin-momentum correlations at RHIC. The uncertainties on the results are a 50-fold improvement with respect to those of the one prior measurement for the same observable, from the Fermilab E704 experiment. These results constrain gluon spin-momentum correlations in transversely polarized protons.