Can diamond undergo a self sustaining combustion reaction all the way to carbon dioxide?
Yes, diamond will combust in air. Regardless of the ambient air temperature, e.g. your example of $21\ \mathrm{^\circ C}$, you of course have to heat it to it's ignition temperature somehow, whether in a furnace, a flame, etc.  The autoignition temperature for diamond is around $900\ \mathrm{^\circ C}$ (source 1, source 2), compared to about $730\ \mathrm{^\circ C}$ for graphite.  The autoignition temperature is described in this Wikipedia article as:  


  "...the lowest temperature at which it spontaneously ignites in normal
  atmosphere without an external source of ignition, such as a flame or
  spark. This temperature is required to supply the activation energy
  needed for combustion."


So, once you have initiated combustion by some means heating it to about $900\ \mathrm{^\circ C}$, it will continue to burn, primarily to $\ce{CO2}$.  Note that combustion does not require that a flame be produced or maintained.

How efficiently diamond combusts in air, in other words how much of the carbon is converted to $\ce{CO2}$ vs. $\ce{CO}$ and other partially oxidized carbon species is a function of a few different things. The same is true when burning graphite, wood, or most anything else. Is it a single smooth "rock" or a finely ground powder? How much of what sort of impurities does it contain? How efficiently ventilated is the system? These issues and other conditions of the combustion will determine the degree to which $\ce{CO}$ and other partially combusted carbon species are produced.