The question may have not been correctly worded. The term "API" was misleading to a couple of folks - and I apologize for that.
What I was trying to do was to "share" a video by linking it in Facebook, AND to have it play in the profile feed (ie. to not leave the Facebook site while viewing the video - just like YouTube and Hulu.com do).
After digging into the JavaScript that Hulu was using to get this behavior, we noticed that a link tag was missing between our version of the <meta>
and <link>
tags, and Hulu's.
Facebook's web page, http://www.facebook.com/share_partners.php, was missing a tag that you should have if the content is a video, and you want it played in the profile feed.
Along with:
<link rel="video_src" href="http://www.example.com/player.swf?video_id=123456789"/> ...
You should include:
<link rel="media:video" href="http://www.example.com/player.swf?video_id=123456789"/>
That will get it to play in the feed. BUT, your domain name must be whitelisted.
To get your domain whitelisted, fill out this form: http://www.facebook.com/share_partners.php#/developers/developer_help.php
I got a reply, and approval, within a day! Kudos to the Facebook developer support staff!