Yes and no. Yes in that it started life as coral and/or coquina. It normally has shells and fossils in it. Mainly it is composed of multiple crystaline forms of calcium carbonate, typically from coral, shells, and plankton. The only reason not to is that it typically contains phosphorous/phosphates in higher levels than ocean soaked rock.
Most farmed live rock starts out as mined coral deposits. The main quaries are Ocala, Miami, and The Bahamas. The rock spends one to three years in the Gulf of Mexico or Florida Bay where it gives off its phosphorous/phosphates. You can do the same with a "rock cooking" procedure. Basically to do that, you keep it in salt water (marine salt) without light and let bacteria consume the organics and phosphates.