I think what you're essentially asking to do is to ask the application that opens the PDF to print, which unless there are proprietary hooks that certain applications use, is not possible (and outside of Adobe, probably won't be very cross-platform friendly).
Javascript's print function is a "standardized" method to tell browsers to print, and so generally that is safe to perform, but both PDF applications and the plugins they provide to browsers so they may display a PDF "in-browser" don't seem to have hooks to do so.
Once the PDF has been opened, that control scheme is outside of your domain of control. As @jorgebg mentioned, either print an HTML page, or let them download the PDF so they can do whatever they please with it. Unfortunately there's no clear way to do both.