Can we consider Cooper as a racist in his book The Last of the Mohicans?

This question is difficult because Cooper's views are outdated in the present day. However, upon examination of the novel in the context it was written, the answer would, most likely, be "no." Remember, that in the nineteenth century, sexual relations between different races was illegal in the United States. However, Cora is presented as a woman of strength and courage—even though she considers herself "a violation of nature" due to her interracial background—and she is willing to give up her life for her sister, Alice.
With that being said, Cooper does portray many of the characters as stock characters such as the "noble savage." More interesting, however, is that Cooper kills off Cora and Uncas (rather than allowing their union), which could be interpreted as his abhorrence for racial mixing. But, on a deeper level, perhaps Cooper intended their death to illustrate and delve into the problematic moral issues facing the new republic—those based upon oppressing indigenous peoples and implementing racial division, which began with the founding fathers.