Do games download okay from the website? We use the same download server on both. If not, then maybe there's a connectivity issue that's blocking requests? Your browser might give you an error message that could tell us more.
Yes, it works fine straight from the website as I did it to double check if that was the issue but still doesn't work?
Does the download eventually fail with an error? Can you post your %APPDATA%/itch/itch.log.txt file somewhere?
Also, which games have you tried downloading with the app so far? Links would be helpful :)