Although you are able to delete your WordPress.com site and the domain mapping upgrade you may have purchased for that site, I would recommend the following instead:
- If you didn't purchase any domain mapping upgrade on your WordPress.com site, go to Settings > Reading in your dashboard, and make your site private. This way, you will keep control over that address if you ever want to use it again someday. If you delete the site, you won't be able to use the site address again. While you may not think of any use for that address at the moment, it may become useful again for another site in a few years, who knows!
- If you did purchase a domain mapping upgrade, start by following the steps I suggested above, and then go to Store > My Upgrades and make sure that Auto Renew is disabled for your domain mapping upgrade; this way, you can simply let the domain mapping upgrade. Until it expires, it will automatically redirect readers who may try to access your old WordPress.com site to the new site.
- If you registered the domain on WordPress.com, you will want to keep the domain there and renew it every year, or transfer it to another registrar as suggested here.
In any case, you can rest assured: changes to your old WordPress.com site won't affect your new self-hosted installation of WordPress.
I hope this helps.