The execution of court orders is done by the executive branch. No human being can even be executed done without the cooperation of the executive.
In the case of Terri, the police power of the executive branch was used to restain people from feeding and hydrating her--in other words, to enforce the unconstitutional court order. A simple executive order could have put that to an end, Law enforcement is under the command of the governor, and failing that, the Commander in Chief.
This is not an execution in the legal sense.