Yes, technically you can arrest a vessel. That is what it is, and has been, for the past couple of hundred years.
The arrest warrant is issued, by a court, then depending on the jurisdiction, a lawyer or bailiff must climb aboard, and arrest the vessel to prevent it leaving the territorial waters of the country where it is under arrest. In some jurisdictions, a company can pay a bond to cover the claim against its vessel to have it released while the court case proceeds. Legally, it is probably the most fascinating area of maritime law, and that's not even getting to sister-ship arrests.
But I'm really not sure how your comment about Miranda rights has anything to do with the arrest of a vessel.
Foreign vessels can be held in port under court orders until creditors are paid for long overdue food and fuel services. Vessels can be seized for legal reasons inside territorial waters for human trafficking, drug smuggling, etc. We have held vessels from sailing until the spill they caused has been cleaned up and a Surety Bond has been posted prior to allowing them to depart.
AlaPlaya is right. Foreign vessels can be seized/held for many reasons. Many foreign vessels have Ship’s Agents in ports to handle these affairs. Legally any agency that seizes a vessel has an obligation to maintain it in good order. That doesn’t always happen. I’ve seen them sit idle and unattended for years, and quickly deteriorate.
The SeaFarer’s Union has legally held vessels to get the vessel owners to pay the crews long overdue wages. This doesn’t always end well. Many times, the vessel owners leave the crew on their own and the Seamen’s Mission has to provide food and fuel to enable the crew to survive during winter. Salvors also have a legal right (liens) to hold vessels in port until all their bills are paid.
The Flying Fox is an expensive toy, located well outside US territorial waters, registered under a flag of convenience. Even with a US-DR agreement in place, I think it would be hard for a US Agency to seize the vessel under the guise of the spoils of war. The Caymans have a reputation as an offshore money haven and secrecy. I’m not sure if they have a reciprocity agreement with the US. I doubt if they would cooperate with US authorities.
Seizures of foreign vessels, even in territorial waters, can come back to bite enforcement agencies in the ass. We seized a Spanish trawler years ago, allegedly for illegally fishing a restricted species in a restricted area. We lost the case and had to pay for all costs including loss of fishing time and loss of the processed fish on board the vessel. It cost our government millions of dollars and serious loss of face as the Japanese say.