Picard needed the Enterprise to remain at Galorndon Core in order to rescue Geordi, whenever one of the brief atmospheric windows of the planet allowed. He also needed to deliver an injured Romulan to Tomalak's warbird in the nearby Neutral Zone. Picard chose to take a hardline approach, remaining at Galorndon Core even knowing that Crusher could not treat the Romulan as well as Romulan doctors. (And, of course, he refused to order Worf to donate whatever blood product it was the Romulan needed to live (though the Romulan soon refused any treatment using Worf's blood, rendering the point moot).) Picard also threatened Tomalak with severe penalties if he crossed into Federation space to retrieve the officer himself. Tomalak soon crosses over anyway, but the Romulan dies while he is en route.
Upon Tomalak's arrival at Galorndon Core, Picard narrowly averts a battle . . . one which he seems to want to avoid. It was sheer dumb luck that Picard was able to do so . . . the window to Geordi opened up and there just so happened to be a second Romulan survivor with him.
My point here is not to question Picard's hardline approach to the Romulans, or his willingness to let the Romulan die instead of leaving Geordi behind on a world that would slowly kill him. The latter was just fine though the combo wasn't quite in character, but whatever. However, the simple fact is that Picard forgot something.
The Enterprise-D had separation capability. This means the saucer section could have remained behind at Galorndon Core, while the stardrive section and the dying Romulan could have travelled to the Neutral Zone to challenge Tomalak.
I don't see any way to avoid the conclusion that this was a big ole plot hole. In-universe, of course, this would've simply been a questionable choice on Picard's part . . . one which almost resulted in a shooting war with the Romulan Empire.
In short, Picard lucked out.
(It's also worth noting that "The Enemy" contains some info useful in the forthcoming Warp Drive Sucks page, in which I detail as many instances of warp velocities as I can find. The entry for "The Enemy" currently reads:
"The Enemy"[TNG3] . . . Tomalak tried to hail the crashed scoutship to let them know that their distress signal had been received. He was entering the Neutral Zone and would be at their location in six hours. When he talked to Picard moments later, Picard revealed that he had intercepted the transmission, and identified Galorndon Core as being half a light-year inside Federation space. Moments later, Tomalak stated that he would be at the Federation border of the Neutral Zone in five hours. That suggests that his ship would have taken an hour to go the extra half light-year to Galorndon Core. (Possibly Tomalak intended to make the trip at the best cloaked speed that would allow him to avoid detection . . . as of 2371 ("The Die is Cast"[DSN3]), the best such speed was warp six.) In any case, the clear velocity in the incident is .5ly/hr, or 4380c.
The above would also suggest that the Neutral Zone was 2.5ly wide near Galorndon Core.
Regarding the plot hole above, it's also worth noting that Picard had shuttles to work with. At least in Voyager, shuttles were pretty consistent in being able to travel above 500c.
"Macrocosm"[VOY3] . . . Janeway and Neelix arrive via shuttle to rendezvous with Voyager, but it isn't there. Janeway uses the shuttle's sensors and detects it adrift over a light-year away. After boarding the ship, they find an abandoned meal that Janeway determines was replicated eleven hours ago, along with Neelix's morning show still playing after having been activated at about the same time. Assuming people weren't happily watching Neelix's morning show and replicating meals while the ship was adrift, then the shuttle must have travelled over a light-year in less than eleven hours. Assuming one light-year and ten hours as the travel time, then the shuttle's velocity must have been .1 ly/hr. That's 2.4 ly/day, or 876c.
"Tsunkatse"[VOY6] . . . Seven and Tuvok intend to take a shuttle to study a "micronebula approximately one point six light years from here on the verge of collapse". Seven notes in the next scene that they anticipate being gone for approximately 48 hours. That requires the shuttle to be capable of travelling no less than 1.6 light-years per day. That's 0.0667ly/hr, or 584c.
It makes sense to presume that Janeway had the pedal closer to the floor in "Macrocosm" than it would've been in "Tsunkatse". At such a speed, Picard could've had the Romulan on a shuttle and at the Federation border of the Neutral Zone in time to meet the warbird upon its arrival, five hours after the first conversation with Tomalak.
Of course, it's possible that the Romulan could've needed medical tools not available on a shuttle at some point during the trip . . . I sure as hell wouldn't want to be piloting the shuttle if the Romulan died en route.
(Hell, I wouldn't want to pilot a shuttle going to meet a warbird even if things were peachy!)