The standard 'Scientific' explanation is that the carbon-carbon bonds in diamond are too stable, no enzyme would be able to overcome the energy barrier necessary to disassemble diamond.

However, diamond is something of a special case for carbon compounds (oh, fullerenes are probably pretty inedible). There are organisms that 'eat' rocks, reduce gold salts to elemental gold and other improbable diets (from the point of veiw of sugar-eaters). There are bacteria that derive energy from sodium and some that produce hydrogen or eat methane. Many organisms can synthesise silica polymers - and I don't in principle see why you couldn't engineer bacteria to make silicon chips.

The message is that it's more surprising what Life can do than what it can't.

Does Thermodynamic Stability Imply Biological Stability?

(If you cannot see the fundamental importance of this you are a cretin.) 

In his famous book 'What is life?' Erwin Schrödinger asked the question 'Why do not organisms eat diamond mush?'  The Alchemist's argument is based on the idea that catalysts cannot overcome large energy barriers - why is that then - it is, at least for me, news?  (Recall that evolution is God-like in its ingenuity.)

If a diamond were thrown into a cess pit it would not be attacked by the microbes, even though it has the energy content roughly of sugar - the diamond has biological stability (persists a long time).

If it were to hang in space it would take a very long time to evaporate - it is thermodynamically stable.

What Schrödinger was asking was, I believe, 'Does thermodynamic stability imply biological stability? and 'Does biological stability imply thermodynamic stability?' Should the answer be 'yes' this would be a profoundly important fundamental of both physics and biology.