Yes, you can! :)
Just go to e3.nintendo.com when the live stream begins and click on the video.
Nintendo changed their Directs to the same format Youtube uses, so normally it works. I watched the last two on my gamepad and I was even able to watch the Xbox One reveal and E3 on it, so it would be weird if it didn't.
And @FrancisNobleman, unfortunatly you can't download the Nintendo Channel anymore in Wii mode, only on normal Wii's...







