Edit: snark aside, this is a valid question. Especially bathroom doors, considering he admits to rushing off to the bathroom to wash his hands and his claim that people don't wash their hands after visiting the bathroom.
Edit 2: I'm starting to feel a little bad for joining in on the Arrington bashing, the guy sounds like he's got some form of OCD neurosis. Though I suppose calling for a ban on handshakes (unthinkable in many cultures) as a result of it strikes me as self-centred.
I wasn't trying to bash Arrington, btw. He has a point about germs. Especially with the sheeple freaking out about Fludor the Burninator and such. But a handshake ultimately is about showing trust (whether rooted in showing lack of a weapon or not), and here he is saying outright that he does not trust anyone else to keep their hands clean to his obviously-high personal standard. I repsect that he's willing to be forthright about it, though I think carrying a travel-sized bottle of Purell might be a solution with milder social consequences in certain business settings.
Incidentally, the door handle in a public bathroom can have significantly more bacteria on them than the toilet seat. So ladies (and gentlemen I guess) when you hover over the seat, reconsider if you actually want to touch that door handle. IIRC one study found a 20:1 ratio between doorhandle and toilet seat (door handles are wiped down with a regular cleaner, whilst toilets get a professional-grade bleaching).