Diamonds ARE minerals.

More Info: Diamonds are made of pure carbon atoms that are arranged in a lattice-like crystal structure.  They are the hardest substance known on earth that occurs naturally boasting a 10 on the Moh's scale.

Why Is a Diamond Classified as a Mineral?

A mineral is a naturally occurring, solid substance.  It is neither vegetable nor animal and is inorganic or is a derivative of an organic substance, such as coal.  A mineral has a definite crystalline structure, chemical composition, color, and hardness.  Diamonds are minerals.

Characteristics of the Mineral Diamond

Scratch Hardness: The diamond is the hardest mineral on Earth, assigned a 10 on the Moh's scale of hardness.  The mineral's hardness does vary between diamonds, but even the least hard diamond is substantially harder than the mineral corundum, which sits at 9 on the Moh's scale. ["Diamonds", Britannica.com]

Anatomical Structure: The diamond's strength is derived from its anatomical structure, the densest of any mineral.  The diamond's structure is composed of carbon.  For every carbon atom within a diamond, there are four additional carbon atoms surrounding each connected by strong covalent bonds. 

Melting Point: The diamond has a melting point of 3,550°C and a boiling point of 4,827°C where the melting point denotes when the mineral will turn to liquid and the boiling point denotes when the mineral will turn to gas.  Not quite as high as carbon or amorphous, but in relation to minerals such as gold is substantially higher. ["World of Carbon", invsee.asu.edu]

Thermal Conductivity: A mineral's ability to transfer heat is known as thermal conductivity.  Though most transparent objects are poor conductors of heat, at 23.2 W/(c*K) the diamond's thermal conductivity is high due to its structure.  This makes the diamond seem cool to the touch. ["World of Carbon", invsee.asu.edu]

Refractive Index: The ability of a substance to bend or refract light is measured by the refractive index.  The ability of an object to refract light depends on the anatomical structure of the atoms and molecules of the object and how tightly the electrons are held by the atoms.  The electrons of most transparent objects such as glass allow light to pass right through, while the diamond has extremely high refractive abilities making it appear to sparkle and shine. ["Refractive Index" Britannica.com]