The term Republican was adopted in 1792 by supporters of Thomas Jefferson, who favoured a decentralized government with limited powers. Although Jefferson’s political philosophy is consistent with the outlook of the modern Republican Party, his faction, which soon became known as the Democratic-Republican Party, ironically evolved by the 1830s into the Democratic Party, the modern Republican Party’s chief rival.
The Republican Party was also called the Democratic-Republican Party. This is all very confusing, but the main thing to bear in mind that when the US constitution was established, it was not contemplating a multi-party system.
Before the establishment of the second political party of the United States, it operated as a one-party state.
Democratic-Republican Party, originally (1792–98) Republican Party, first opposition political party in the United States. Organized in 1792 as the Republican Party, its members held power nationally between 1801 and 1825. It was the direct antecedent of the present Democratic Party.
The Republican Party was founded in 1792 by Thomas Jefferson to protect agrarian interests from the centralising tendencies of Alexander Hamilton. Democrats split from this party in the 1820s over tariffs during the presidency of Andrew Jackson. The splittist Democrats supported Jackson.
Before the American Civil War ideological disagreements were about whether federal power should take over the rights of individual states and if so, how this was to be done. The American Civil War was also about this. The Southern states just wouldn't see why they couldn't carry on practising slavery if they wanted to because it wasn't any business of the North's, and were prepared to fight for this principle. They lost. But that is another story.
The point is that there is nothing stopping the US from becoming a one-party state, like China. Even if it did, the operation of the 1787 constitution would continue to apply.
The only thing that would change is that the only party in the United States would send competing candidates from the same party.
Daniel Bell has written a book on how the US might benefit from taking a leaf out of the book of the Chinese Communist Party. http://press.princeton.edu/titles/10418.html
The trouble with the US is that its politicians are weak and stupid, as Paul Craig Roberts would attest, and therefore vulnerable to the machinations of factions whose interests are not necessarily in the interests of the American people eg feminists, Zionists and the military industrial complex. You can be sure that if the Chinese adopted representative democracy they would soon also be having a "Jewish problem" too. Representative democracies do what the Koran warns against: dividing their political ideology into sects and rejoicing in their own doctrines. http://www.internetmosque.net/read/english_translation_of_the_quran_meaning/23/53/index.htm
When urged to accept representative democracy, the Chinese simply opined that it was too divisive and would militate against good government. Could they be right?
In any case the US constitution when drafted never envisaged a multi-party system.