No, it is false that one immediate outcome of the Battle of Thermopylae was that Athens and Sparta finally united, since this battle resulted in a Persian Victory.
The United States presidential election of 1876 was one of the most disputed presidential elections in American history. Samuel J. Tilden of New York outpolled Ohio's Rutherford B. Hayes in the popular vote, and had 184 electoral votes to Hayes' 165, with 20 votes uncounted.
The North and the South had contrasting views on slave labor while the Federalists and Republicans fought on how much power the government should have. Each of these groups believed in America, and thought that their viewpoints would make it stronger. Their opinions began to contrast so much, that eventually a civil war broke out.