You have to ask why the coroner's office was not called, and an immediate test done of his blood.
Why was the mortuary called? Are they going to do a quick burial?
Lord have mercy on him, and grant him eternal rest.
Having some acquaintanceship with death care procedures, I presume the coroner WAS CALLED; he certainly would have been if this had occurred in WA State. It would be unusual for a news story to indicate the presence of the coroner unless there was something sensational about the circumstances, but the coroner always goes to the place of death or to where the body is taken, depending on the circumstances.
I'm assuming this is a rural area. Usually only cities of some size will have a public morgue. In MOST areas a funeral home is called to remove the deceased and to shelter the remains in a refrigerated unit until the coroner releases the body to the family. In MOST areas if the coroner deems an autopsy necessary an outside pathologist (a Medical Doctor) needs to be brought in and the post mortem examination is done at the funeral home. The cost to the county for an autopsy can be $2,000 to $5,000; travel expenses and perhaps overnight lodging for the pathologist, his professional fee, a fee to the mortuary for using their facilities, outside lab work, etc. The necessity of incurring these costs is often a point of contention between County Commissioners and the local Coroner (an elected office in WA excepted in the 3 or 4 largest counties which have a Medical Examiner hired by the Board of Supervisors). Coroner's want to often do autopsies to eliminate any speculation, County Commissioners would rather have as few as possible because of the cost.
In this case, as the decedent wasn't a local resident and one presumes there was no evidence of foul play at the death scene, the county (I'm presuming that county commissioners in rural Texas are a pretty conservative bunch) would be MORE than happy to get the body out of town before it cost them any money. The death certificate for Justice Scalia would be issued by the local coroner and recorded in the Texas public record, the exact procedure varying in each state.
If the family decides to have an autopsy. or if having one would be standard protocol because of his position, a delay of a day or a few should not impact the accuracy of lab work. I'm only guessing, but it is something of an educated guess, that an autopsy would be performed at the Walter Reed National Military Medical Center, then the remains released to the funeral home of the family's choice, which I would guess to be Joseph Gawler's Sons in Washington, DC.