It is important to remember that the response to European arrival varied by location and culture. In the Southeast we see Spaniards arriving to mixed reactions, but they all really fit into the existing interactions between people living in these areas. Some groups tried to use the Europeans for their own benefits, others wanted nothing to do with them. To us the interactions appear to be New World and Old World, but to the Native Americans the Europeans were just another group moving into the area. A great account of these interactions is De Soto's exploration of the Southeast. He actually ended up meeting some of the late remnants of Mississippian cultures, but what is more relevant is his interactions with the cultures of the Southeast. To keep it brief, De Soto and his men were kind of jerks the entire time. Most of the conflicts that he faced appear to have been initiated in some way by someone involved in the expedition. They also had this tendency to force natives into guiding them or serving as translators. So they were not the most popular bunch in the area. One modern take on these interactions was that it was the ground work for future European interactions and approach to cultures in the SE. I do not personally feel that is completely accurate, but it probably did impact things to an extent.

The reason I mentioned that is to hopefully show how much miscommunication occured during many of these initial contacts. Both sides would have been trying to angle things to get the better deal from the interaction. Word would have spread that new people were in the area, we see that De Soto tried to play the "I am your god" card for a few groups he met. This would have all been part of why the Spaniards often had harsh welcomes in this region, they were not behaving in a manner that would make anyone want to allow them in the area. On the other hand, some of De Soto's guides were Spaniards who were living as full-fledged members of these communities. So the foreign aspect was probably not the cause of the hostility but instead the actions of De Soto.

So to answer your question, any response was tied to the actions of the Europeans themselves. The Native Americans responded to the Europeans as they responded to any other group moving past them. They may have warned other communities, but I doubt it was a response to them being Europeans and more just a "hey some guys moved into this area and are being kinda rude and dangerous, lets kick them out." This was a response they probably had had prior to the arrival of Europeans and not unique to European arrival.