It wouldn't be the "External" storage that would be full but the "Internal" storage. Apps run off and use the "Internal" storage. "External" storage would be a SD card.
The only thing you can do is clear off space on the device to make more room for it. If you have a SD card, then move things like photos, files, music, etc., (not apps) to the card to free up space. Or you can also uninstall any apps you don't really use or clear up the data on some apps. Like not keeping every text message you ever get, or every email, the history to every site visited, etc., Clearing up that stuff will also free up space.