The Doctor was a specially designed program and wouldn't necessarily know everything about Starfleet's first-contact missions or military engagements.
By rights, the Doctor shouldn't know anything
outside of medicine, and what he does know should be concentrated in emergency medicine at that.
I disagree. As long as there's sufficient memory capacity (and there is), the EMH should have as much knowledge as they can possibly give him. The more he knows about Starfleet and its operations, the better he can treat patients who may have been hurt during those operations.
It's impossible to know TOO much.