Does sensor downscaling necessarily improve resolution?

The issue is not pixel density (i.e., pixel pitch). It is the total pixels in the image that determines the likelihood of moire. If you take a similar image, the one with more pixels will be less likely to show moire than the one with less pixels, even though the image with less pixels has a higher pixel density (i.e., a smaller pixel pitch).

You can confirm this in a few seconds by comparing the Pentax K5IIs and D800E samples on Imaging Resource. The pixel density is slightly higher on the K5IIs, yet the K5IIs shows moire where the D800E does not.