No, Gallery isn't a place, so you can't download "to" Gallery. You can download, then either restart or run media.Re.Scan, so the downloaded picture gets added to the media database, then upload it to Facebook. (If you leave it in the Downloads folder, though, it gets deleted after about a month, so put it somewhere else if you want to keep it [use Copy, then Paste, then delete the original - .if you use Move, and something goes wrong during the move, you could lost the picture].)
What if you just reboot after you've downloaded the image, as Rukbat also suggested? Also, use a file manager app to make sure there's no file called .nomedia in the directory where the downloaded images are saved.