Secret police is right. How does he know who McCoy is anyway? Has he run his ID through a database off camera?
As a member of the Enterprise's senior staff, McCoy was connected to the "Genesis Issue" and therefore a familiar face to this Fed arresting him.
I'm telling you, Kirk's career was over when he left the Enterprise
considering the mess in ST II, but nobody including Morrow had told him, yet:
- Ceti Alpha V Disaster: Had Kirk failed to report what he did in "Space Seed" or had a Starfleet employee screwed up big time? It must have been Kirk's responsibility to check whether the system was safe to abandon Federation citizens there (interesting legal question concerning Khan & company). Apparently it wasn't and practically Kirk had condamned them to death.
- Loss of USS Reliant: Captain Terrell (and Chekov!) were obviously unaware that Ceti Alpha V had been inhabited by Khan & company (well, they would have checked, wouldn't they?). The loss of NCC-1864 was therefore due to the original filing mistake and Captain Terrell could no longer answer for the loss of his ship. Probably Kirk would have become the scapegoat to blame the loss for.
- Violation of General Order 12: Resulted in fatal injuries aboad USS Enterprise and substantial damage of NCC-1701 because of Kirk's negligence. Log records show that Kirk didn't listen to advice of navigational officer.
- Mutara Sector losses: Regula lab scientists, engineering crew of USS Reliant, several crew members of USS Enterprise, USS Reliant
Inevitably there would have been a board of inquiry. I'd say it didn't look too good for Kirk and his career would probably have been over - only in the real world, of course.