I have integrated the YouTube API only on the index page of the website. It works in all major browsers, including IE 7 and 8. I have tested it on Iphone's, Ipod Touch, Ipad 1st Gen, Ipad 2nd Gen and on various Samsung devices. The actually video from the API does not display on any of these devices (it's just blank space with the CSS styles. The video list "More Videos" does display and appears to function. I cannot figure what is causing this problem?
If you just grab the standard YouTube embed code (the iframe code is fine) it works nicely on all devices. They way you are doing it now seems just to serve up the Flash version, which i-Devices don't like.
To get the gallery-like effect you want, there are quite a few slideshow scripts out there into which you could paste the standard iframe code.
But perhaps the API discusses this problem and what to do for i-Devices? I'm not sure. I did a bit of Googling, and found discussions on it, but they are a bit over my head.