As this is a murder suspect, the procedure becomes very simple. It's harder to figure out something like a DUI with serious injuries to the passenger. Usually in that case, the police arrest and release, because then the government isn't going to be on the hook for the medical bills. And the police can easily keep track of the patient and if they want custody when he's released, that's easily dealt with.
But a murder suspect? The government is probably going to want serious bail even for something like second degree murder. So the police arrest the suspect at the hospital or scene, and he's now in their custody. The Constitution allows the police to hold for a certain period without a bail hearing. Effectively by playing around with this a bit, you can get between 48 - 72 hours of hold without a bail hearing, ignoring holidays and weekends. At this point, the suspect is in police custody, not correctional custody. The police can restrict access to the suspect, and this includes family and clergy. Even an attorney can have problems getting access to his client during this period, without the Constitution being violated.
Once this hold period expires, the police need to hold a bail hearing. At that point, a judge will come to the hospital and hold a bail hearing. This will meet the Constitutional requirements even if the suspect is in a coma. Usually for murder suspects, bail is either not available or beyond the ability of most mortals to raise. Custody of the suspect is transferred to the corrections department. At that point, the police stop guarding the inmate and correctional officers assume the duty. Visitation is allowed in accordance with the corrections department visitation policy, which would include clergy certified to be members of the clergy.
This is all Constitutional. On security arrangements, the hospital director will defer to the government. Security is the responsibility of the police and/or corrections department and the nurse can allow whoever she wants to visit. They still ain't getting in.
Realize this is done in a very low key fashion. ICU is very easy to control, and the hospital will already have visitation rules. But even in the regular rooms of the hospital, you can usually position this stuff for privacy. And as IronMike points out, at least in the past, D.C.'s General Hospital had a lockup. I doubt when it got closed it was discontinued. My guess it just moved somewhere else.
Jim Clark-Dawe