An important issue is that the American President is elected separately from the determination of the majority party in the legislative body in the United States.
In most European countries, the head of government is the leader of either the majority party in the legislative body (House of Commons, L'Assemblee, Bundestag, etc.), or more likely, the head of the majority coalition, in a multiparty setting. In this kind of situation, it makes sense for a number of small parties to form in order to represent competing interests in the legislature, then band together to choose the national leader. Under "European" rules, the President of the United States might be Nancy Pelosi, House Majority Leader, and we, like Germany, would have had a female CEO already.
But in the U.S., the race for the Presidency is distinct from the legislative election process discussed above. It has its own special rules, including the Electoral College alluded to in the last paragraph, which cause the Presidential race to lend itself better to a two party system. Basically, a President needs to win with a plurality in each state, and then win a majority of state electoral votes. In 1860, Abraham Lincoln won with about 60% of 60%, that is a majority of votes in states with a majority of electoral votes. His theoretical vote total could have been 36% or even less, but he won a few "stray" votes elsewhere to come to 40%.
"Third party" candidates (Ross Perot in 1992, Robert LaFollette in 1920) tend not to gain traction. (One partial exception, the "Bull Moose" Party, was led by a former President against the incumbent President.) Although this is true mainly at the Presidential level, this has also discouraged fragmentation of the legislature as well.
Here's how "European" rules might have worked in an American context. In the 2000 Presidential election, George Bush Jr. and Albert Gore Jr. both won 48% of the popular vote. Ralph Nader, of the Green Party won 3%. Assuming that legislative seats had been apportioned in proportion to the popular vote, Nader, with his 3%, would have formed a coalition with either Bush or Gore (probably the latter), to arrive at 51%. Conceivably a winning "bidder" (possibly Bush), might have made Nader president in exchange for the Vice-Presidency and a dominant Cabinet. (In 1933 Germany, conservatives made Adolf Hitler Chancellor in exchange for the Vice-Chancellorship and 8 out of 11 Cabinet seats; they thought they could control Hitler, and they were wrong.)
In 2000, nothing of the sort happened. In fact, Bush was made President by the electoral college even though he had fewer popular votes than Gore. Not only did he have fewer votes than Gore plus Nader, he even had fewer votes than Gore and Nader together in Florida (the "tipping point" state).