If you really developed and "finished your Windows application" and even "created setup file", I guess, you need to send it to the customer, to run it. :-)
But as you are in doubt, I would advise to make SQL server (
SQL Express and all compatible versions of all SQL Server products) as one of prerequisites; and your installer can check it up. It depends in what installation toolkit you are using. Just for example, you can read how to do it with
WiX (highly recommended):
http://www.advancedinstaller.com/forums/viewtopic.php?f=2&t=24127,
http://wix.tramontana.co.hu/tutorial/sql/creating-a-database,
http://sourceforge.net/p/wix/mailman/message/27011065.
Why not deploying SQL with your installation? Probably you can (check up your license and the rights for redistribution), but in this case better make it optional installation. Why? Because licensing is a complicated issue; and because the user need the opportunity to use the SQL server products they already have, or have a license, and it could be different from what you use. Generally, installations should be sometimes tested on a fresh computer. You can use a virtual machine for such purposes.
—SA