He could, but he won't and most likely won't shut it down at all.
While campaigning, candidates make a lot of promises they can't fulfill once in office. Why does this happen? Because they do not know all the details of the situation, they do not have the highly secured information that the president and his advisors are privy to.
Once in office, the president is briefed and gets the whole picture, at which time, he may realize that he cannot change the situation or, at least, change it with the ease he thought he could.
In this case Obama was looking at it from the human rights prospective. Guantanamo Bay detention camp has become a black mark on American Foreign Policy.
Once he realized that it is very necessary to keep it open for National Security reasons; he realized that it would be VERRY difficult to close down and VERY unwise.
I am quite sure you have noticed that this administration has backed down rather quickly and quietly about this topic. It is clear they hope that people will forget he ever made a promise to close it in the first place.