I recently came across Web Intents, specifically Twitter's.
I was looking for (and couldn't find) if Facebook has such API, specifically for posting to groups (public groups) and posting to some (public) wall.
Yes. https://developers.facebook.com/docs/sharing/reference/share-dialog
https://www.facebook.com/dialog/share?
app_id=145634995501895
&display=page
&href=https%3A%2F%2Fdevelopers.facebook.com%2Fdocs%2F&redirect_uri=https%3A%2F%2Fdevelopers.facebook.com%2Ftools%2Fexplorer
Facebook supports sharing intent: http://facebook.com/sharer.php?u= (read this) and dialog feed: https://www.facebook.com/dialog/feed? (documented here and here - how to change its popup size). While Facebook doesn't document the sharer currently, I've never seen any notice that it is going to be deprecated at any time and many websites use it.
On our website I use both the sharer (when the shared object has a page associated with it - sharer.php relies on OG tags, you can test any page at Facebook debugger) and the dialog feed (when the object doesn't have a page and I need more control of picture/other elements of the share). The main difference of dialog feed, that can be its disadvantage, is that the object is shared via your app, so it is less likely to appear in user's friends' feeds. Also the picture preview shared via dialog feed is smaller. The object shared via sharer.php looks and behaves as it would if the user shared it, that in many cases could be a preferred way.
It looks like the (new?) sharing dialog has URL support and allows sharing not just URLs but open graph events as well. It is the closest thing I've found
https://developers.facebook.com/docs/sharing/reference/share-dialog
Short answer: No. There is no such thing as an RESTful API, per se, that you can implement easily that will take care of interaction/authentication for you.
That being said, here are a number of social plugins which seem to do what you are looking for: https://developers.facebook.com/docs/plugins/