Why is U used for ICs on circuit diagrams?

U was originally the designator for "unspecified". I suppose that made sense when integrated circuits were new. There was no existing category for them, so they went into the none-of-the-above category. I use "IC" as a designator for ICs because I think it makes more sense, especially nowadays when ICs are common.

Good question - "U" stands for:

inseparable assembly
integrated-circuit package
microcircuit
micromodule
photon-coupled isolator  

I think the U was picked because these are all classed as "Unrepairable", since they are all in sealed packages.
FWIW, the designator for separable assemblies is "A"

Note that it was actually "IC" (and other things - for instance I've seen Z used) before it changed to U. If you look at schematics from pre ~1980 you will see a lot of different (to todays standard) designators used, e.g. CR or V for diode, G or X for crystal, etc