People were already adjusting the formula for APS-C bodies, saying it needed a * 1.6... or even a * 2 due to the high number of pixels compared to early DLSRs or what was acceptable in film. So yeah, if you need sharpness at the pixel level for a large print, or due to a need for an extreme crop, or whatever, then yes, you need a higher shutter speed.

But as said above, you don't always need that extra detail to be perfectly sharp. You can usually downsize. It's a case by case thing.

And yes, technically it's not because the sensor pixels are small, but because of how little of the image each pixel covers. For the sake of not confusing people though, it's easier to just talk about high megapixel counts or pixel density.