Yes, they can turn down voluntary leave of absence. They did it to me. They can't mess with FMLA though. You need a note from the sick person's doctor that describes your part in their recovery (work will give you paperwork). You have to have been there for so long to qualify for it though. Memory is foggy. Not too sure what the req was. A year, I think.
Although you have your answer, your question seems to say "well I'm not taking a paycheck, so why can't they just give me leave like I want?"
A company hires "x" number of bodies to do "x" amount of work. Your request for leave messes with that balance. Depending on the size of the company, there may or may not be "slop" in the picture. A mom and pop shop with three employees can't afford to have one employee say I won't be here for a year. Why did they hire you in the first place?
A 5000 person company may have a set percentage of "slots" for sick/leave/maternity/military/etc. But let's say they plan on 1% or 50 employees gone at any time - and you come along and ask to be the 1.02 (51st employee) to leave. If they let you go, they now have to hire someone to fill your slot. But you are asking them to hold a place for you when you come back - even though they are holding 50 other slots for other folks.
The answer may come back as "no". Stay or resign. But your request (just by the nature of being behind 50 other people) is the one that will break the camels back.