How does the rule of law of the ancient Greeks resemble that of modern America? Ancient Greece may have been the birthplace of democracy, but how did their version contrast with that of modern America?

Ancient Athens instituted a form of democracy in which one male citizen equalled one vote. There were few enough male citizens (much of the population was female or enslaved) that they could all assemble in a central square and vote with their voices on issues at hand.
This system was implemented to fight corruption among the ruling class. Too many laws were being instituted that were destructive to the rest of society. They were to the advantage only of the very small ruling group. Democracy was meant to provide greater protection to less powerful citizens, who were increasingly becoming impoverished by laws instituted from above.
We can easily see of the differences between this form of democracy and what we have in the U.S. We have universal suffrage, which means that all citizens (at least those who are not felons) can, in theory, vote. We also have too large a population to assemble everyone in one place to voice vote on issues. Instead, we elect representatives who are supposed to carry out the will of the group of people who voted for them.
The closest we have to Athenian democracy is the referendum. In a referendum, if in an issue garners enough support through signatures, it goes on a ballot and every citizen in a state or a jurisdiction can vote on it. In this case, the side of the issue that gets the most votes becomes law. For example, states or cities might have referendums on raising taxes or raising the minimum wage. If enough citizens vote to raise taxes or wages, that decision becomes the law.
No democracy is perfect but the purest democracies include and value the voices of as many citizens as possible.

Ancient Greek democracy was both more exclusive and more participatory than modern American democracy. As in the modern United States, the citizens of Classical Athens enjoyed equality under the law. As in the early United States, Athenian slaves and women were not considered citizens and had no political rights or representation. There was typically little recognition of individual human rights in the ancient Greek city-states; on the contrary, civic patriotism demanded that citizens sacrifice their lives and pay voluntary monetary contributions to maintain and safeguard their homeland. During the heyday of Classical Greece in 5th century B.C.E., a citizen militia constituted the backbone of Greek armies.
Radical Greek democracies, such as Athens under Pericles, stressed political engagement of poor citizens and provided these citizens with various opportunities to influence government policies. Poor Athenians participated actively in the courts and the popular assembly. The Athenians elected most officials by lot. This largely eliminated corruption and prevented moneyed interests from controlling electoral policies.