Turbulence induced by Rayleigh-Taylor instability is a ubiquitous phenomenon with applications ranging from atmospheric physics and geophysics to supernova explosions and plasma confinement fusion. Despite its fundamental character, a phenomenological theory has been proposed only recently and several predictions are untested. In this Rapid Communication we confirm spatiotemporal predictions of the theory by means of direct numerical simulations at high resolution and we extend the phenomenology to take into account intermittency effects. We show that scaling exponents are indistinguishable from those of Navier-Stokes turbulence at comparable Reynolds number, a result in support of the universality of turbulence with respect to the forcing mechanism. We also show that the time dependence of Rayleigh, Reynolds, and Nusselt numbers realizes the Kraichnan scaling regime associated with the ultimate state of thermal convection.