Is it possible to get a grass block using a dirt-only Minecraft world and naturally spawning mobs?
There is a Minecraft world with only dirt.
Difficulty is normal mode.
Is it possible to only use mobs that naturally spawn and this dirt to get grass to spawn?
No, I don't think there is any way to grow grass except for using grass next to it. It's all according to this: http://minecraft.gamepedia.com/Grass_Block