Yes, wealthy people do have an obligation to give to charity, if they believe in a higher power that is. The Bible says that the the “higher power” sets up kingdoms and tears them down. Therefore, a wealthy person should realize where the wealth comes from and give to some less fortunate people. Wealthy people who believe in self-made success most likely will not feel obligated to help anyone, however.
Realistically, they really don't. Morally, I think there is some obligation to help. I'd like to believe that those who are so incredibly wealthy would think of others who are less fortunate and support them through charitable donations. In J.K. Rowling's case, she's been on the dole and knows what it's like to worry about money and basic necessities, so I love that she's generous.
Generosity is an attribute that must be measured at the individual level. A person of great wealth can be very generous while another person who lives in poverty can be just as generous. To say that the wealthy have a moral obligation to donate is wrong. We all have a moral obligation as human beings to donate whenever possible to help others less fortunate than ourselves, whether through monetary donations or acts of service.
No, charitable donations can never be an obligation. They have already paid their taxes like everyone else. In a progressive tax system like ours, that means they already paid a lot more than the rest of us. Imposing an obligation to donate money beyond that amounts to invoking a second, unofficial tax on moral grounds.