World's oldest democracy
The claim the United States is the oldest democracy is usually made by US nationalists. The argument consists of two points, that the US is a democracy, and that it is the oldest of world democracies. Plolitifact accepts this argument (under specific definitions), but I disagree with both points of the argument.
The Oldest In What Way?
To say that something is the "oldest" is an ambiguous term. For example, the "oldest" person could be the oldest person still alive today, the person in history who achieved the longest lifespan, or the first person to be considered a Homo sapiens.
Since city-states in ancient Greece appear to be the first governments to have democracy, one of the definitions for "oldest" is not true. To remove such ambiguity, the claim should be reworded to, "the US is the oldest extant democracy." But this brings up the second point.
Is the United States a Democracy?
I think most sociologists would describe the government of the United States as a constitutional republic rather than a direct democracy. The difference being, in a republic, people elect officials to handle political matters for them while in a direct democracy, every citizen has an equal vote in every political matter at all levels of government.
However, since no country is a direct democracy at all levels, it is fair to say that the US is similar to a democracy in the sense that it has regular elections to choose and alter the government where all adults can freely and fairly vote. However, the US wasn't always this way. At the time it identified as an independent nation, the only people who could vote were white men, a fraction of the population.
In my opinion, a nation is not a democracy unless it has universal suffrage, that is, the freedom for all adults to vote. The US did not adopt universal suffrage until 1965, the 38th nation to do so. However, it was one of the first to allow a large amount of the general population to vote in meaningful elections.