Honestly seems blown out of proportion on both sides.
For the medics, just stop wearing the damned socks. Or wear higher boots.
For the service, find a softball way of handling this. Winning on the sock issue at the cost of general morale is not a winning solution. Heck in the long run it might be easier and cheaper (though hard to prove that one tangibly) to buy and issue black tube socks.
For example, my service requires a navy or black t-shirt as an undershirt. Somewhere along the line they started issuing service logo t-shirts to fill this role. I didn't like them. But I didn't start wearing whatever t-shirt suited me, I bought black undershirts. Were I to wear the wrong colour, the service could always point to the issued shirts and show me my options.