I'm not sure that there's a clear understanding of how this technology works, and to say that you won't accept a solution if it requires any expense on your part is going to strongly limit your acceptable solutions.
Since you listed your only hardware as being your desktop computer with a wired ethernet connection, the data lightning-to-USB cable, and your iPhone, then the setup might imply that you want to use your desktop computer for either ICS or a network bridge. But, actually, I have this sneaking suspicion that you are talking about something entirely different which is USB reverse tethering. If you need further clarification on a network bridge or ICS, I could discuss it later but I don't think either will deliver the result that you're looking for. And, as I said before, what I believe you are asking for is USB reverse tethering. It's not a feature that either Android or iOS supports without root.
But, if you changed your mind about purchasing additional equipment, you could set up a wired ethernet connection without rooting your device. You would need the lightning-to-USB cable (which you have), the power adapter for your phone (which you have), a USB-to-ethernet adapter, and the Lightning-to-USB 3 Camera Adapter or a compatible powered USB hub.
Otherwise, it's really not possible to share an ethernet connection with your computer because your phone without root can not perform USB reverse tethering. It's only made possible through additional equipment because of the adapters, and the power-over-ethernet that is made available by connecting all of those adapters.
See also: Required parts for connecting ethernet to phone
Just accept that this isn't going to happen – Steve Jobs – 2017-05-31T04:16:00.123