It's established that the Doctor is on the holodeck and running a simulation. Lieutenant Barclay says he needs to destroy the holographic imaging system to confirm his being a real person. However even when the Doctor does do this he's still on the holodeck and therefore he wouldn't have disappeared because he destroyed the simulated holographic imaging system. Had he destroyed the real one in Engineering he would've indeed disappeared. This is clearly a false way to prove to the Doctor that he's real.
When the Doctor is in the sick bay with Janeway, Nelix and the Kazon he gives the computer a command to shut down all holographic projections. Janeway and co disappear, however the infrastructure of the ship remain, even though those features are also holographic projections.
The ship computer tells the doctor the sick bay does not have any holograph emitters, then a bit later, Cpt. Jane way walks in, as a hologram simulation.
When the Doctor is shown the personnel file for himself as Lewis Zimmerman, the included image shows Zimmerman wearing a uniform with a gold-colored shoulder yoke, indicating operations/engineering division. While the real Zimmerman is in engineering, at this point in the simulation, the computer was trying to convince the Doctor that he was the human chief medical officer of Voyager named Lewis Zimmerman, not the real Lewis Zimmerman back on Jupiter Station. As a medical officer, the Zimmerman in the personnel file should have been wearing a blue-colored uniform.
Like Tom Paris in Parallax (1995), Barclay speaks of "the Voyager" instead "Voyager". Voyager is ordinarily designated without the definite article.
After Barclay tells the Doctor he needs to destroy the ship the Doctor says, "First of all, I don't know how to destroy the ship. Second of all, I wouldn't do it even if I could, and third: How do I know you're not an alien trying to trick me into killing everyone?"
Barclay responds that these are good questions. Only the last one was a question, the first two were statements.