What role accretion disks play in the formation of high-mass stars remains a long-standing question. For example, the accretion rates in massive star formation may be high enough to induce gravitational instabilities and put disk-mediated accretion in doubt. The few observations reporting disk-like accretion flows around high-mass protostars have not had the spectral and spatial resolution to measure the properties of the candidate disks. In this talk, I will present new millimetre-wave spectral line observations of the CH3CN molecule in the accretion disk around the massive protostar IRAS 20126+4104 that for the first time measure the disk density, temperature, and rotational velocity with sufficient resolution to assess the gravitational stability of the disk as a function of radius through the disk. The Toomre-Q parameter summarizes the competing effects of the forces affecting the dynamical stability of the accretion flow: the differential shear in the rotating disk and the gas temperature together stabilize the disk against the clumping tendency of the self-gravity of the disk surface density. Our observations find that in contrast to some theoretical expectations, the disk is stable at all radii which permits a smooth accretion flow. Our results put forward the first constraints on gravitational instabilities in massive protostellar disks, which are closely connected to the formation of companion stars and planetary systems by fragmentation.