Not sure you ever really had a true democracy... not sure any country does. Especially not the ones who actually call themselves 'Democratic republic of... whatever' as it always strikes me that they are trying to prove too much in the name and never do it in action.
America is (and always has been) a republic with some democracy in the way that Britain is a monarchy with some democracy. Both have an elite, select group who effectively control it all. The fact that a president can effectively over rule the vote of the senate and do what he damn well pleases is a sign of this - no different in a fundamental way to various early modern English monarchs (Charles I as a prime example) who randomly decided to dissolve parliament and rule without their support. If one individual (whether he or she is elected or not) can eliminate one aspect of the democracy from the system by effectively ignoring it then that is not a democracy.
As I said above, I am not sure the ideal of true democracy exists in practise at present. I think that the size of a population is important in democracy - on a small scale it works really well but after a certain size it begins to lose power. You need things to support it - parliaments, senates, congress - and they bring with them a whole host of problems which detract from true democracy.