Sendmail “Connection refused by mail.example.com” when attempting to send emails

I'm trying to have a virtually hosted PHP site send emails to a particular Gmail address when the site's contact form is correctly filled out and submitted. Unfortunately, nothing is being sent out. In /var/log/mail.log I have numerous instances of things like this:

Jul 31 16:38:51 picus sm-mta[28576]: v6SL4xII018234: to=<person@gmail.com>, delay=2+23:33:52, xdelay=00:00:00, mailer=esmtp, pri=38910659, relay=mail.example.com., dsn=4.0.0,stat=Deferred:Connection refused by mail.example.com.</person@gmail.com>

Where person@gmail.com is the address the site-generated email is to be sent to, and example.com is my domain name. Note: I don't have a reverse DNS setup for this domain because it's virtual.

I'm completely new to handling an email server, and simply wanted to use Sendmail since it's built into the distro I'm using and Swiftmailer supports it, so I'm not sure how to fix the problem. Any ideas?