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You may want to show a different icon with different viewport sizes. For example, compare the different sizes of the Google Calendar icon here:

http://setup.googleapps.com/Home/user-resources/google-icons...

You'll note the 16px icon uses square corners, while the other sizes have rounded corners.

There are other considerations as well, such as causing unintentional Moiré effects, and scaling DPIs in not-quite-powers-of-two (basically all of Android).

Still doesn't solve the design-problem of showing e.g. a differently-cropped (or entirely different) image to different viewports.

Doesn't that mean I would be downloading all of the image data and only using part of it?

No. Part of the http spec is the ability to download only a specific range of bytes in a file.