What an excellent question... although not exactly a historical question unless you mean: "Has any state officially declared itself to be the Fourth Rome, or could any state or persons be considered to have made some analogous claim." There is a book called "Moscow, the Fourth Rome" which relates the idea to the USSR and Eisenstein's film Ivan the Terrible in which one character says something like: "There have been three Romes. The first two have fallen, but the third (Moscow) will never fall. And there will be no fourth!" One could also relate Albert Speer's plan for Nazi Berlin to a "fourth-Rome" ideology.

Of course... in a broader sense there are, and have been, many Romes. Rome itself is composed out of multiple layers, added by successive powers in the course of various attempts to concretize political prestige, from the classical period onwards. Furthermore, Rome has been repeatedly invoked throughout modernity as a source of legitimacy, from the French Revolution to the architecture of Washington DC. I'd guess the USA would today be the most plausible candidate, although I would personally nominate Skopje, "Macedonia."