Turns out the answer to this problem can be found in GDK, not GTK.
The following code is what I've managed to figure out:
GdkDisplay *display = gdk_display_get_default();
// 0 for the first screen, as we're only concerned about the first one
GdkScreen *screen = gdk_display_get_screen(display, 0);
screen_width = gdk_screen_get_width(screen);
screen_height = gdk_screen_get_height(screen);
Note that you'll need to use the gdk_display_get_n_screens(display)
function call to determine the number of screens if you're interested in any screen other than the first one.
For reference, check out the documentation for GdkDisplay, GdkScreen, and multi-heading.