I've not seen the film, but I'm curious as to whether one factor was that Tonto was being played by a white actor (a move even the 1950s serialization didn't make). A quick glance at the reviews on Rotten Tomatoes doesn't show this leaping out as an issue, but while Depp is famously versatile I imagine there would be outrage if he were to play a well-known African American character - shall we say, Jim from Huckleberry Finn? (There might well be outrage if there were to be a remake of Huckleberry Finn at all, of course.) Then it occurred to me to wonder whether the actor who plays Jacob in the Twilight series is Native American (he's not). I guess that giving Native American roles to white actors is just not seen as problematic - but I'm puzzled as to why. I'd have thought that a history of genocide (as of slavery) would make a crucial difference in these cases, these days.
By contrast, I sigh a little when I see that the villain in The Lone Ranger is yet again played by an Englishman, but while there is no doubt an element of racism involved it's mostly just a creakingly unimaginative cliché.