Michael already gave a short answer, but I'll be a bit more pedantic.
Wouldn't be an issue if iOS opened up a little. Real file system argument time again....it also give swap file access as a bene. The little section of the drive anything can use. Running app A and B split they currently get their little world and whatever ram is free to access. Break down those walls, app can see oh I have gb's of swap file to play with now.
Swap is managed by the kernel. It has to be, since it is tied into the memory management interactions between the processor's virtual memory map, and the kernel. Apps don't even have their own swap files on the desktop unless they are simply so bloated, they decided it was a good idea to override the OS' RAM management and try to invent their own poor alternative.
That said, apps do have their own space on the disk to play with. It's their sandbox. They
can put whatever they want there. If a painting app wants to store layers on disk as a way of keeping their memory usage down, they are totally free to do so. You do trade off some performance to do it, and it isn't exactly cheap in development time to write such a beast for the first time, but it can be done.
The decision to not use swap on iOS was a design decision made by Apple. I don't know the exact reason, but I can easily imagine a couple reasons why it wasn't implemented:
1) The swap file would wind up on NAND. SSDs with many, many NAND chips have to use wear leveling to extend their lifespan of writes. Apple does something similar (the file system encryption keys are actually kept in a part of NAND that is immune to wear leveling), but it only has a couple chips instead of the many an SSD uses. I suspect the lifespan isn't quite up to the same level as a good laptop/desktop SSD, and a swap file will simply thrash and decrease the lifespan considerably.
2) Storing the contents of RAM on NAND is a bit of a security risk. The ability to push sensitive information into a swap file isn't exactly a new attack either. By avoiding a swap file, you avoid having to mitigate that sort of attack.
And Apple has been pushing at ways to allow apps to manage their own memory without a swap file. They notify when memory gets low, so that you can flush data to disk if you need to and release the RAM. iOS will even force background apps to quit to reclaim memory. Something not even OS X has at the moment. I could certainly leverage this mechanism for saving more complex state to disk. Many apps do not do this, sadly. This is likely another reason they haven't considered adding a swap file yet, because there are other approaches being employed to keep RAM available as applications request it.