How to determine if a user has not changed default generated password

The following will tell you if the user has updated the password generated by WP on registration for a user with ID $user_id:

if ( ! get_user_option('default_password_nag', $user_id) ) {
    //then the user has changed the default password
} else {
    //the default password has not been changed
}

If you want to send an email @goldenapples’ answer is good (replacing the logic in user_password_check for checking whether the password has been changed with my logic.)