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?
'91 Toyota Pickup
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"They say the test of literary power is whether a man can write an inscription. I say, 'Can he name a kitten?'" -- Samuel Butler