All known official Chao desktop wallpapers are listed here for you to download. If you know of any more, please let us know. To download a wallpaper, first decide what resolution you need; you probably want the largest one. Then either right-click the link for that resolution and choose "Save target as..." or "Save link as...", or left-click the link, let it load, and then right-click the image and choose "Save image as..." or "Set as wallpaper".
This was released by Sega in the Sonic 20th anniversary app for the Apple iPhone. Various wallpapers were available for download for a limited period of two weeks only!