Is there a systemwide .bash_profile file?
The answer seems to be yes.

Check out the FILE section of Bash Man page

FILES
   /bin/bash
          The bash executable
   /etc/profile
          The systemwide initialization file, executed for login shells
   ~/.bash_profile
          The personal initialization file, executed for login shells
   ~/.bashrc
          The individual per-interactive-shell startup file
   ~/.bash_logout
          The individual login shell cleanup file, executed when a login shell exits
   ~/.inputrc
          Individual readline initialization file


In addition to that, shell paths are also loaded from /etc/paths and the files (if any) in /etc/paths.d by /usr/libexec/path_helper (which is executed as part of /etc/profile).

Lastly, there's also /etc/bashrc, intended for functions and aliases (while /etc/profile is designated for system wide environment and startup programs).