The Romulan was a prisoner of war, if not officially. Crusher's first duty is to preserve the life of her patients, and especially that of a prisoner. Starfleet does not kill its prisoners.
He was neither a prisoner nor a "prisoner of war" as the Romulans and Federation were not at war. (At most they were in a "cold war.")
The Enterprise found a crashed Romulan scout ship on a Federation planet and rescued, presumably, its only crew member. He had severe injuries probably exasperated by the conditions on the planet on his physiology. He needed some kind of transfusion (not strictly a blood transfusion but some other body greeble made up for the show) to survive his injuries. Crusher tested Enterprise crew members to find a match and turned out Worf was the only match. (He did not have a problem with being tested.) Crusher tried to enlist Worf's donation but he refused citing personal/cultural disagreements with the Romulans. The Romulan seemed to hold similar biases.
The Romulan was never considered a captor.
Communications with an incoming Warbird revealed tensions between Picard and the Romulan Commander. Picard refused to allow the Warbird to enter Federation space so that they could come and retrieve their fallen comrade. At the same time Picard refused to bring the injured Romulan to the TNZ so the Warbird could retrieve him because Geordi was still lost on the planet and they were awaiting in a window for sensors to find him/transporters to work.
The Romulan Warbird eventually broke The Neutral Zone and got to the planet where a confrontation (mostly chest puffing) occurred between Picard and the Romulan Commander. By this time the injured Romulan had succumbed to his injuries.
(All of this time there's question on why the Romulan scout ship had violated The Neutral Zone treaty and how many crew members were on the crashed ship.)
Meanwhile on the planet Geordi meets another survivor of the crashed ship. Both are suffering the effects of the planet's radiation/magnetic field. It's made Geordi's VISOR useless to him as visual aid and it's made the Romulan weak. There's some initial tensions between the two (The Romulan posturing that Geordi is "his prisoner" and will be captured when the Romulans return) but eventually they overcome their differences to find a solution to get off the planet. Geordi is able to direct the Romulan on how to combine the VISOR with a tricorder to get it to pick up the signal of a nearby probe to use it as a homing signal to the Enterprise.
The Enterprise receives the homing signal and in act of "trust" to the Romulans drops the shields and beams Geordi and the crash survivor to the Enterprise. Picard assures the Romulan he will not be harmed. The surviving Romulan reports to his commander that he has revealed no information, and has survived due to the help of Geordi. Picard then releases the Romulan to Romulan Commander and, presumably, escorts them back to TNZ.
Again, I think it's really a good episode to show the contrasts of the way different people in the episode treat "the enemy." On the one hand we have Worf refusing to help the injured Romulan survive as well as the injured Romulan refusing treatment. We have the posturing between Picard and the Romulan commander over protocols and the treaties keeping a semblance of "peace" between the two peoples.
And then we have Geordi and the Romulan on the planet working through their differences and working together to solve their mutual problem. There's even a nice moment where the Romulan seems surprised Geordi's parents "let him live" due to his disability. But it ends up being that disability (namely the prosthetic that allows him to live through that disability) that helps save the both of them.
As I said above the biggest problem I have is that Picard seems a bit needlessly irrational in the episode. Given the information he has, that the injured Romulan was the only man on the crashed ship, it seems like it would be in his and the Federation's best interests for that man to survive lest it make more problems with the Romulans.
It's illogical that Picard refused to allow the Romulan ship to come sooner so they could treat their injured man or that Picard didn't take the ship to TNZ to deliver the injured Romulan. He was putting the life of ONE crew member on the planet over the possibility of higher tensions or war with the Romulans. Picard's a smart man and he should have seen this.
As I said up-thread he could have allowed the Romulans to come get their man provided during the trip they take a specified route and stayed uncloaked. At that time the "excuse" by the Romulans was that the crashed ship crossed TNZ into Federation space due to a navigational error. Very likely a lie, and Picard knew this, but he could have given the benefit of the doubt and dealt with that issue later.
His #1 priority in that situation should have made sure that the Romulan he was housing lived and was given every opportunity to live. If his personal ethics, or the ethics of Starfleet, preclude him "ordering" someone to undergo a medical procedure against their wishes then his hands are tied. There's only one place that Romulan can get the treatment he needs and that's on a Romulan ship. Picard should have allowed the Romulan to get there. If he had to abandon Geordi on the planet in order to take the Romulan to TNZ then that's the risk Geordi took when he signed up for Starfleet.
If he has to allow one Romulan ship in this very special circumstance to cross TNZ then so be it.
But, again, that was all sort of the point of the episode. The different ways we treat "the enemy." And in all of it, clearly, Geordi and the planetside Romulan survivor handled it the best.