First off, if we drop the big about neural networks, we already do exactly as you say. We have software which takes high level instructions and generates highly optimized low level code (all the way down to the binary machine code) which humans could never possibly do on their own. On modern large projects, it is simply unreasonable for a human hand to write all that machine language code at the same level of speed a modern compiler can.
So what changes when we add neural networks into the mix? The first thing that changes is that we no longer have any sort of guarantee that the program we request will actually be what we get. Neural networks don't offer that kind of guarantee, and they are never designed to offer that guarantee. They're designed to learn and find the patterns you didn't think to tell them about. If you only ever use a neural network to do what you told it to do, you're wasting everyone's time.
So what you're looking for is not just some technology. You're looking for us to desire the development of software to fuzzy standards. There needs to be value in providing an incomplete story of the software we want to see, rather than providing a complete one. Find a reason for why we would want this sort of technology, and it will appear.
One possible reason: unique user experiences. If each user wants a slightly customized experience, they may want their device to learn their habits. Maybe one user has a slight twitch which causes some single-taps to become double-taps, so they want double-taps to be a bit harder to accidentally do. To program to this environment means programming to ten-thousand environments. That's hard to do, and very fuzzy. A neural network may be able to take your code (written in some high level form) and information about how the user wants the program to act (gathered by their own neural network), and custom write an application for that user and their UI style.
Of course, you may already have a fuzzy task in mind. In FPGA design, there is a notoriously nebulous step called "routing" where the compiler figures out where to put elements on the chip. This process is already being explored as a candidate for neural networks. So, in that sense, your question of "is it possible" can indeed be answered with "yes, it is possible, and we do it already."