I'm surprised Icahn is allowed to have a twitter. Regulators used to limit access to certain information sharing outlets for those in positions of influence, e.g. certain high level bank official could not have BBM (I know right) active on their Blackberries because of the risk of insider trading. When a tweet has the power to move the market it should probably be treated the same.
Information that moves the market is allowed, as long as all investors have access to the information.
There was some contention about that last year with the NetFlix CEO releasing subscriber numbers on Twitter that affected the stock, but the SEC issued guidance earlier this year that said that disclosure on Twitter and Facebook were ok, as long as the accounts were public.
I think if you really try even your account could move a stock. Think about it: A tweet with location (in front of a big company's CEO's house) early in the morning when you know the 'target' is commuting, flying or otherwise offline, followed by a tweet 'just killed the bastard' with a fake photoshopped picture of a body'. If someone was to target a person with a love of guns on their timeline... well things could get messy real quick.