Right now it is only possible to restrict pages behind a password to people who are logged in, members, control panel users or admins.
It would be nice to be able to put other pages behind a password for people who have a certain permission level, membership level (so not just any member) or, even a tag.
This isn't planned but it's something that we can definitely see the value in. If you have more information on how this would help your organization please let us know below!
UPDATE: See Brian's workaround solution in the comments below.