I was interested in how the size of the new universal USB-C connector compared with Apple's existing Lightning connector, so I looked up the dimensions and drew up the graphic below. I also included a standard USB type A port for reference - you can see that USB-C and Lightning are quite similar in size.
I was also interested to see whether Apple might use USB-C in the next generation of iPhone, especially considering new rumours regarding the removal of the headphone jack, so I have included an overlay over an iPhone 6 wireframe.
As 9to5mac point out in this article (where they link this graphic), the USB-C port is about the same height as the headphone jack, meaning little would be gained by switching to a USB-C only solution.
This could mean that USB-C might remain Mac specific, and Lightning might remain iDevice specific - which would make sense in order to avoid breaking compatibility with the entire ecosystem of Lightning accessories.
Edit 05/11/2018: The 2018 iPad Pro now comes equipped with a USB-C port. This means the chances of an iPhone eventually adopting USB-C have increased significantly.