Indeed, all memory management in iOS is virtual, they just don't use a pagefile on any iOS devices. If the pager's not built in already and just turned off, I'm sure it'd be a very simple thing to add.
But then it would have to swap, the user experience would suffer unpredictably, Steve Jobs would strangle the engineer responsible and the feature would never see the light of day.
"Virtual memory" does not imply "it would have to swap"; indeed, the iPhone does have virtual memory, and each application has its own virtual address space.
As others have pointed out iOS does have virtual memory. As to why it doesn't use a swap file probably the a reason is that flash memory has a limited number of read-write cycles, probably way more than you could ever use under normal usage but with paging stuff to and from flash all the time you might reach the limits much faster.
That may be a factor, but I think it's much more likely to be related to interface fluidity. People swap to SSDs on desktops/laptops without significant issue.
I believe Android uses a similar scheme: virtual memory but no pagefile. Is there a way to turn on paging on Android?