You can, but it is a PITA. Just above the "Submit" and "Preview" buttons" is a line,
"Also share this as an update to" and the Facebook icon.
It is checked. Uncheck it.
With Mozilla Firefox running under Windows 7, (and probably many more OS/Browser combinations) you have to uncheck it EVERY time you post a question or answer. (Not when you vote for a BA, though.) With other operating systems and browsers, it stays unchecked for a session, or at least some other users have reported it staying unchecked.