Yes, you can create a custom workshop (let's call it "the medical backoffice"), and a custom reaction that doesn't produce anything (let's call it "paperwork"). That would produce what you are talking about, and if you made The Medical Backoffice only accept the doctor as a worker, you could have them perform paperwork constantly so long as some patient wasn't walking in the door.
This shouldn't really need to be necessary, however - if you just make sure that all of a doctor's labors besides being a doctor are turned off, any time they aren't spending on medicine will be spent in the nearest meeting room, which you could simply have relatively near the hospital, and they won't do other jobs (except maybe harvesting crops). Burrows don't block off access to anything, they simply restrict what jobs a dwarf will take - dwarves can path through areas not in their burrow to reach jobs that are in their burrow.
Common diseases were part of the goals before, when 31.01 was coming out, but was stripped from the actual release, and Toady will probably want to revisit it at some point to put it back in, but that time hasn't come so far.
Workplace injuries were suggested before, but reactions are mixed to it - occasional toe-stubbing or bashing one's hand with a hammer is one thing, but people tripping and falling off of cliffs or down stairs and dying for no particular reason is a major potential annoyance, especially if the diplomat trips into the moat and drowns, and suddenly, you're at war for letting the diplomat die.
Speaking of which, since you are new, it's worth pointing out that we heavily encourage people to use the search function in the top right of your screen to try to look for similar ideas that have been suggested before, and to read the reactions to those before you post your own ideas. Rather than rehash the same arguments over and over again, it's better to learn from the arguments of the past, and try to take the best of the old, and correct the old mistakes.
If your suggestion is almost identical to an existing suggestion, it's probably best to just bump the old one and add onto it, rather than recrating it all over again.
It's not that this thread was a common suggestion (I've only seen the workplace injury suggestion once or twice before), but it's a problem that many new posters run into, so it's worth telling them about it.