The war that you mention here did not have any particular name. It is not known as the Spanish-Indian War or anything like that. Instead, historians tend to speak of two major campaigns. They talk about Cortes's conquest of the Aztecs and they speak of Pizarro's conquest of the Inca. They do not speak of the Spanish-Aztec or Spanish-Inca War.
One thing that should be noted about these wars is that they were not necessarily against all of the indigenous peoples of the Americas. Instead, the Spanish tended to use Indian allies to help them defeat the powerful kingdoms. Both the Aztecs and the Inca had many enemies who were happy to see them fall. Of course, the Spanish came to rule these allies as well, but there was no overall war between the Spanish on the one side and all of the Native Americans on the other.
Thank you phonpel
This clears out many thoughts and questions.