The major debate sparked by this war had to do with whether non-whites could ever really sustain a democratic system and whether including them in the American polity would be harmful to America.
This was an argument on two fronts. First, if places like the Philippines were taken as territories of the US, would the US be harmed? On this point, people argued about whether having the Filipinos as American subjects would harm American democracy. Some believed that the Filipinos, as non-whites, could not participate properly in a democracy and would become a problem for the country just like (as people thought then) the African Americans in the South were.
Second, how much autonomy (or even independence) should such places be given? People in the US tended to believe that non-whites were simply not capable of ruling themselves in a democratic system. Therefore, they felt that the US should not give much autonomy to the Filipinos.
The Spanish-American War, then, caused a debate over how compatible non-white people were with the responsibilities of citizenship in a democracy.