The only way to hide it is to use a PO Box as your FCC mailing address. As a FCC licensee your contact information is public for a few reasons, like any other TV or radio station. It's also public for your convenience, so that other hams know where to send QSL cards. I don't know that it's a major problem unless you're telling patients your call sign and even then do you suppose they will they know where to look?
"Stand your ground. Don't fire unless fired upon, but if they mean to have a war, let it begin here." -- Capt. John Parker