Has SpaceX showed any interest in building satellites?
...
You could argue that it already does build large satellites, specifically Falcon 9 second stages!

Seriously though, satellite development is a very different business and technical arena and unless Elon has thought of some hitherto unexplored way to build large-scale commercial satellites at dramatically reduced cost, I can see that it would be of little value and yet more distraction to an already very busy R&D roadmap.
If anything, I would think Elon would start a separate sister company but as I said, I doubt it; too much else to do.