Specifically what I'm looking for is the embedded video player functionality you would receive if you had just manually shared a YouTube video but done programmatically through the Graph API.
3 Answers
It appears that you have to extract the URLs of the actual swf in the page and the thumbnail image yourself.
For example, this seems to work:
curl -F 'access_token=...' \
-F 'message=Link to YouTube' \
-F 'link=http://www.youtube.com/watch?v=3aICB2mUu2k' \
-F 'source=http://www.youtube.com/v/3aICB2mUu2k' \
-F 'picture=http://img.youtube.com/vi/3aICB2mUu2k/0.jpg' \
https://graph.facebook.com/me/feed
See source: Posting an embedded video link using the Facebook Graph API

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You can either use the old REST API or the new Graph API. In either case you must have the publish_stream permission.
If you want to use the Graph API you need to create a Post object and then post it to the graph feed url for that profile
graph.facebook.com/PROFILE_ID/feed
You will want to set the source property in the Post object to the youTube url for the video.
http://developers.facebook.com/docs/reference/api/post/
If you want to use the older REST API the process is similar but you post to
api-video.facebook.com/restserver.php

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Facebook does not allow youtube to play video on facebook anymore but direct them to youtube website to watch.

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