In theory, diamond growing is possible
Diamonds are famously hard, but when you get down to the molecular level, it's nothing particularly impressive - just a carbon with four carbon bonds. The standout is that a diamond consistently has these kinds of bonds across the entirety of the diamond. Now, a carbon-carbon bond is very easy to do - the body uses enzymes to do it all the time. All you would have to do it keep building a lattice of these carbons over and over - it's something the body can do. There's no reason an extremophile can't make one.
That said, it's a waste of energy and resources. To make this, you'll need to take a carbon bonded to something which isn't a carbon, and turn that into a C-C bond, which isn't easy to do (read: endergonic reaction) and it's very not easy if it's, say a C-H bond. Something like a C-O bond, or even better, a C-X bond, X being a halide of some kind. And it's a complex process. To make artificial diamonds, they take carbon, and they subject it to vasts amount of heat and pressure - to the point where it was the biggest explosion on Mythbuster's history. That's the kind of energy we're talking about - it's not something to toss around idly.
(P.S. - That clip is a bunch of Mythbuster explosions, followed by the mother of all Mythbuster explosions. You want to watch it.)