It is possible, but only with Facebook Authorization:
https://developers.facebook.com/docs/facebook-login/
The user has to authorize your app with the "publish_actions" permission, and you can post stuff on his wall with the PHP SDK:
https://github.com/facebook/facebook-php-sdk
Just keep in mind that the message always has to be 100% user generated. So, in order to share something like this, you would need to share without the "message" parameter, or you would have to include an input field where the user could set his own message.
Btw, for the functionality you described, there is a much better solution: https://developers.facebook.com/docs/plugins/like-button/