Evidently not, since projecting a simple hologram outside of sickbay would have been trivially easy for even the crew of Voyager to achieve. Simple holograms probably wouldn't work for the doctors program, though, since not being able to touch or interact with anything possibly means he wouldn't be able to SEE anything either (that's just the way his program works; the computer isn't feeding inputs to his program from the ship's internal sensors, it's reading the inputs from the part of his projection that makes up his "eyes." The mobile emitter does the same thing. This is apparently so the doctor can operate on patients with precision and not have to rely on sensor inputs from the other side of the room).
They have the ability to project an IMAGE anywhere on the ship. But if the Doctor was just an image, NOTHING about his character would make sense.
Put it more simply: The Doctor is not actually an AI in the normal sense of the word. The Doctor is a simulated person, essentially a modified holodeck character modeled after one or more real physicians in Starfleet. His program and memories can't be copied or transferred and is remarkably hardware-specific in that it can't be saved to more than one location at once without corrupting the entire program. A simple medical AI or expert system is something WE can already do right here in the 21st century, and projecting an avatar of it on the wall or on a TV screen isn't far off either. But the Doctor is something very VERY different.