It's simply a case of taking the "Doctor is a person" thing too far for the sake of creating a nominal amount of drama. It would make sense that Voyager would have some sort of offline backup for all basic Starfleet-provided data on the system. Transporters, bridge-consoles, the Doctor... everything.
If it did not, then what would happen if the storage medium (isolinear, gel-pac, whatever) got damaged, or infected by an alien compute virus (I mean, really did we learn nothing watching Independence Day

) ?
You could then rebuild or replace the damaged system physically, but without a backup you would need to re-write the software that it runs it, or forensically recover the data and copy it over (which might not even be possible).
Secondly, as far as transporting the Doctor is concerned, I can buy that you might be able to set up a "connection" with the Prometheus over this network, and you might even be able to transfer him over it, but the job of the computer at that point is to read the data and send it. To delete his program, as Kahless implied, the system would need to actively write data over it, and there's simply no reason to do that. The crew would have to be intentionally trying to delete all traces in order to lose him.
Less drastically, even when you move a file from one location to another on your computer, you are copying it, then deleting the old one by removing references to it in the file system portion of your code to reclaim the space. All that data still exists until its overwritten sometime in the future, it's just a little harder to get to.
This is all, of course, assuming computer storage in Star Trek is anything like computer storage today. Then the whole thing fails on 3 levels -
there's no reason to delete him
even if you did, there's not necessarily no hope of getting him back.
Voyager HAS to have backups of essential software, or someone like Harry Kim has to have committed all the code to run those systems to memory (and more or less perfectly so)