Which country traces its roots back to the oldest civilization?

The Sumerians are widely credited with being the first genuine civilisation on Earth, beginning gradually around the 5th millennium BC and reaching clear 'civilisation' status by the early 4th millennium BC. Towns and large agricultural communities existed before this time, but are generally not considered to have constituted a civilisation. The Sumerians, who were situated in modern-day lower Iraq and Kuwait, are widely believed to have invented the city-state and organised government, law, writing, systematic agriculture, the wheel (probably), and irrigation, among various other early innovations and marks of civilisation.

Although the dates of the beginnings of these civilisations are in general not terribly well-defined, the order of the appearance of "civilisations" is often considered to be the following.

• Ancient Sumerian — the very earliest (first city-states in world in 4th millennium BC, cuneiform script, probably first 'proper' writing system, c. 3,000 BC)

• Ancient Egyptian — widely considered independent in origin (gradual development of civilisation throughout 4th millennium BC, unification / start of First Dynasty c. 3150 BC, appearance of hieroglyphs c. 3,000 BC)

• Ancient Indian (Indus Valley Civilisation) — widely considered independent in origin (c. 3,300 BC, Indus 'Script' from around 3,300 BC but may well have just been pictograms for initial phase, since not yet deciphered, syncretism with Indo-Aryans from early 2nd millennium BC)

• Ancient Chinese — widely considered independent in origin (c. 1,900 BC with Ertilou period and development of state, beginning of Bronze Age, Oracle Bone Script recorded from c. 1,250 BC but definitely older, yet still pictographic at this point)

• Ancient Greek (Minoan / Mycenaean) — partially inheriting from Mesopotamian and Egyptian developments, older than the above if you count the Minoans of Crete (c. 3,000 or early 3rd millennium BC, undeciphered Linear A script from c. 2,500 BC, Mycenaean Greece and deciphered Linear B script from c. 1,600 BC)

• Phoenician / Canaanite (Lebanon area) — partially inheriting from Mesopotamian and Egyptian civilisation (first recorded in 15th C BC by Egyptians, but city-states likely several centuries older)

• Ancient Italian (Etruscan / Roman) — partially inheriting from the Greeks (Etruscans and city-states from c. 900 BC, Rome founded in c. 753 BC, Republic in 509 BC, dominant Mediterranean power by late 3rd C BC and defeat of Carthage)

• Ancient Iranian (Medean / Persian) — partially inheriting from Elamite and Mesopotamian civilisations (Medean Empire from c. 678 BC but with scarce archaeological remains, extremely influential and historically important Achaemenid Empire from 550 BC)

Sources for these dates can be found here and here. Admittedly, I miss out a few civilisations like the Elamites, Akkadians, Hittites, and other Anatolian and Middle Eastern ones (and thus perhaps I do them injustice) — that said, these would usually be considered to have left behind a far less marked cultural or historical impact on their respective regions, or indeed even a lesser archaeological or documental history in some cases. The Elamites are perhaps most deserving of a place in the above list, and they would date from c. 3,000 BC, putting them in '4th place'. I believe it's debatable how much they were influenced by Sumer.

Note also that I am restricting this answer to Old World civilisations, since although the Norte Chico civilisation of South America can be considered very early, it does not bear some of the hallmarks of what we typically consider — rightly or wrongly — a 'genuine' civilisation (e.g., a mature writing system, the wheel).