Regarding 'culturally extremely difficult'… YES! It's one thing to have people agree that it's a good idea, but it's an entirely different challenge to get them to do it. I think the first step is to have the hosp mktg/comm team know who the active social media doctors are (they often don't) and then figure out how to engage them. Some are doctors that are too busy to bother, but there will be others that are extremely savvy and coordinated about their soc media strategy. Then supply that latter group with important content (like CDC data and tweets, or links to key content and landing pages.)
And to support your point, I had a conversation with a physician the other day who complained about how the hospital will readily post news about a bake sale but won't retweet their latest journal publication or symposium presentation. So the cultural obstacles are both common and need to be addressed.
Also, in our experience, 1/3 to 1/2 of a given physician's followers are fellow clinicians, which is THE critical demographic for referrals and those all-important US News votes. Which means getting a coordinated strategy with your doctors should be a priority.