I was talking with my parents about the pre-Columbus American civilizations (Maya, Aztec, Inca) and my father asked, "How unique is it that these civilizations disappeared?" I argued with him about whether the use of the term "disappeared" was fair - after all, the Spanish showed up and kind of killed everybody, and to me, "disappeared" implies a lack of foreign influence. This started us on a slightly tangential discussion, which goes something like this:
We talk about Indian and Chinese civilization as stretching over thousands of years, including changes of kingdoms and dynasties, successful foreign invasions, etc. Chinese civilization continued despite the Mongol invasion, for example. Is it because the culture, political structure, and language weren't greatly affected? Is that why we can definitively say, "the Incan civilization ended here?" How similar is this to, say, the Islamic conquest of Egypt, which had its own culture, political structure, and language, that was (I assume) absorbed by Islamic culture?
I'm not really sure where I'm going with this, but basically, are there other cases (and is it valid to say this about the American civilizations) where one civilization is completely deleted, either by someone else or some sort of self-destruction?
You're touching on a very difficult and controversial debate in the historical community: collapse. The entire notion is fraught with issues and ambiguity. Mayan archaeologist Arthur Demrest phrased it well:
Recent discussions of the collapse of civilizations have demonstrated that terminological ambiguity creates much of the controversy regarding comparative issues, including differences in the interpretation of specific cultural historical episodes. The meanings of terms such as ‘collapse’ and ‘decline’ are far from obvious. Furthermore, ambiguity about what precisely is ‘collapsing’ (e.g. ‘civilization,’ ‘state,’ ‘kingdoms’, ‘tradition,’ ‘society’) generates more disagreement than do problems of historical or archaeological interpretation.
Depending on who you ask, collapse for any particular period and in any particular region may or may not have occurred. Even the well-known collapses, such as those of the Maya or the western Roman Empire are reasonably controversial to this day.
However, the other component of your question has a rather different response. I would personally argue that the conquests of the Americas were fundamentally different from the famous conquests of the old world, but this is not to suggest that old world civilizations are the same as their historical counterparts. The latter view would be very much revisionist. Much of what we consider about cultural continuity are influenced by the records we have of the events. In the case of China, the idea of cultural continuity is a bedrock concept at the core of Chinese identity. Until abolishment of the office after the Chinese Revolution, the Emperors of China all claimed the Mandate of Heaven as their claim to legitimate rule. This concept established the continuity of the Chinese nation from the Qin through the Qing. Even the Mongols claimed the Mandate of Heaven during the Yuan period and continued many of the familiar social institutions of previous dynasties.
Similarly, the issue of cultural continuity is an intensely political issue in India even today. However, I'd rather defer discussion of continuity in India to someone more familiar with the details of Indian history. Indian cultural continuity is a highly, highly politicized topic that I'm not prepared to address.
Let me bring this back to the original point: History is quite fuzzy. You can certainly talk about the Inca Empire ending with the execution of Túpac Amaru and in some sense you'd be right. But it would be ridiculous to pretend that in that instant all the Incans suddenly became Spanish. The process of conquest was more gradual than it might first appear and attaching an end point necessarily requires some statement of precisely what is ending. This idea applies to each and every civilization you named. The descendants of the people still exist into the present day. I would not personally say the Inca "disappeared". The people that lived in the former empire mostly continued living, albeit under a different power than before. But it's also not fair to pretend that things were the same as before. The Spanish brought sweeping social changes almost everywhere they conquered. Social hierarchies changed, the labor force expanded, and the economy was revamped to suit Spanish needs. Entire societies were shifted and rearranged in the wake of the conquest. So in another sense, it is entirely fair to say that some aspects of the societies were different before and after conquest.
^1 Demarest, Arthur A., and Don S. Rice. The terminal classic in the Maya lowlands: Collapse, transition, and transformation. Univ Pr of Colorado, 2005.
It's not so much that the ebb and flow of cultures in the precontact Americas are unique, but, especially within the United States, but Native cultures are framed in the persistent meta-narrative of the "Vanishing Indian."
In regards to the Inca Empire, it flourished from 1438 to 1533. Since it was overrun by the Spanish, it a post-contact society as well as precontact.
Túpac Amaru II led a resistance movement against the Spanish and was finally defeated and captured in 1781. The Inca Empire arose from Quechua society, which absolutely did not disappear. There are 7.5 million to 11 million Quechua people today. Cuzco still features a great deal of precontact architecture.
Cultural continuity and revivals could be emphasized, but instead pop culture/mass media tends to reify the "Vanishing Indian" meta-narrative.