You don't necessarily need heat and pressure to make diamonds, and those are not usually things that we biological squidgey things use to grow things like horns, nails and scales.

I think there are two ways currently used for creating artificial diamonds on an industrial scale, one being the one you mention ([High-pressure, high-temperature][1] synthetic diamonds).

But the more interesting one is [chemical vapor deposition][2]. Now, this takes place in a vacuum and blahblah. So it, too, is not really a very good way of growing scales.

But at its root, it's basically just accretion. Which is how most biological stuff grows.

Is it likely? No, it seems like it'd be a lot of work for "diamond perfection", when "good enough" scales are way easier to grow, and tougher (less brittle).

Is it feasible that somewhere in the universe, some carbon-based life-form has developed a way to perfectly accrete carbon atoms onto each other? Sure. And maybe there's a spider-like thing which spins nanotubes.

As protective scales, diamonds would leave a lot to be desired, though. They fracture rather than flexing. When they break, they can split or shatter, rather than merely bending, wearing, or chipping. So that would need to be addressed, whether with logic, handwaving, or lampshading. See https://worldbuilding.stackexchange.com/questions/40532/is-diamond-armor-better-than-traditional-armor/40547#40547 and possibly https://worldbuilding.stackexchange.com/questions/146732/is-a-diamond-sword-feasible/146763#146763 for more info on that part, though.

Another possible limit is the availability of carbon in the body, which is covered here: https://worldbuilding.stackexchange.com/questions/98455/what-size-would-a-diamond-made-from-a-human-be

  [1]: https://en.wikipedia.org/wiki/Synthetic_diamond#High_pressure,_high_temperature
  [2]: https://en.wikipedia.org/wiki/Chemical_vapor_deposition.