Russia is the only country on the list with a non-zero possibility of the restoration of a monarchy claiming spiritual descent from the Roman emperors. Peter the Great was awarded the title of Emperor of All Russia "in the manner of the Roman Senate." The Holy See is the only other state that has a reasonable claim, but the Pope was never significant secular leader in the late Empire and the thought of calling a small fortress in Rome the Heir to Rome is disturbing.
The modern Hellenic Republic and the Republic of Turkey both descend from the remains of the Ottoman Empire, which destroyed the last extant remnant of the Roman Empire at Constantinople in 1453. The line of Kings of Italy follows from Odoacer (not genealogically), who overthrew the last Western Roman Emperor in 410 AD. The Empire of Germany, of which the FRG can be considered a distant descendant, stems from the North German Confederation and the Kings of Prussia, who in turn are ultimately descendants of the Grand Masters of the Teutonic Order, and not from the Holy Roman Emperors, who held to best claim to the title of "Heir to Rome" in Europe. The United States has only some sort of spiritual founding myth of Roman influence shoehorned into its history. Of these four, Germany likely holds the best claim to the title, but it is tenuous at best.
Of course, the actual present-day Heir to Rome is the Marble King under the Golden Gate.