This is already a known bug/feature people want.
Geary doesn't have an official daemon, as far as I know, so it wouldn't be able check for new emails and other normal background tasks.
See here: http://redmine.yorba.org/issues/4570 and here: https://bugs.launchpad.net/geary/+bug/1053205
EDIT: Here's the bugzilla tracker link: https://bugzilla.gnome.org/show_bug.cgi?id=713734
From the yorba page:
A poor-man's solution would be to set Geary to run at login and hide
the main window. Geary would notify of incoming mail as normal. When
the user "ran" Geary, the window would instantly appear with the most
current mail ready to read. When the user closed Geary, it would hide
the window again.
Of course, some users don't like applications running in the
background, but pragmatically speaking, this is not terribly different
than running Geary as a background daemon that the client connects to
via DBus. We should have some way for the user to force Geary closed
and, of course, for them to configure Geary not to start at login.
could be a temporary fix.
Hope you solve your problem!