Taiwan should rightfully be considered a province of the People's Republic of China. At the end of World War II Japan was required to surrender all seized territories back to their rightful owners, although this didn't include Taiwan, which was taken by them in the first Sino-Japanese War, it was written into the treaty. From then 'til 1949, Taiwan was a part of China, and not considered a separate nation. When the KMT finally collapsed that year and the Communists took control of the mainland, the remnants of the government fled to the island of Taiwan and sought the protection of the United States, which was given. Until his death in the 1970s Chang Kai-Shek never gave up hope that one day the Nationalists would reclaim the mainland. Today, there is a growing number of people (Mostly the youth) in Taiwan that support (The "controversial" president being one) the independence movement, but you must realize there are other that the generation that is growing up now, following the death of Chiang Kai-Shek was never taught to really believe that Taiwan was a part of China, however, they were taught to view the Communists as not a good thing. As a result, they feel that because of the different ideologies that Taiwan would be better off as their own. The wrath of China will not allow this and would lead to an all-out war.
In short, China was fighting a civil war, it is most certainly not in any way, shape, or form similar the the American Revolution (Okay maybe it was in the sense that it was a sort of civil war). Let's put it this way, if we had Puerto Rico during the American Civil War, and the remnants of the Confederate government fled there and became sheltered by another country and attempted to declare its independence, would the United States agree to that? NO. Of course it wouldn't! What country would like a portion of it chopped off by an "exiled" government that was sticking its nose up at them?
* Taiwan was not booted off the Security Council...they walked out, moments before the vote.