Yes it is possible. However it is impossible to do at standard temperatures and pressures like that at SATP. It has been done by the Z-Machine at the Sandia National Laboratories by using a pressure of 10 million times normal Atmospheric pressure. To create the pressure, the machine's magnetic fields hurled small plates at the diamond at 34 kilometers per second (21 miles per second), or faster than the Earth orbits the Sun.

Unfortunately in open air, the fact that is it large network of covalently bonded carbons make it extremely hard to melt in the first place. Secondly, diamond at high temperatures will not melt, rather it will prefer to burn, as characteristic of all carbon allotropes. A diamond will burn or oxidize when exposed to a hot flame in the presence of oxygen, for example an oxygen torch with a temperature of 800 degrees C (1,472 degrees F), according to The Merck Index, a standard chemistry reference work.

but it can be melted in the absence of air (or in vacuum) at a very very very high temperature

Since the melting point of diamond, the highest melting point required of any mineral -- 3820 degrees Kelvin -- a melted diamond is uncommon.

Read more about the mineral diamond, below.

Not humanly possible, but if temperatures reach (I will get back to you) it is possible.

It is pressure dependent. A normal atmospheric pressure it sublimates at about 4,330 degrees Celsius.

Yes, but only at extremely high temperatures, something like above 2000 degrees Celsius.