No they don't. This challenge wasn't based on someone else having rights to the name, and nobody gets the "Redskins" trademark now that the team has lost it. No property (physical, intellectual, electronic, etc.) has changed hands, and this decision doesn't set precedent for that to happen in the future.
If the Apache group of tribes claimed that the word "Apache" was offensive, and managed to convince the USPTO that that was the case, maybe they could get the trademark rescinded, but that wouldn't mean they would get it for themselves, nor would it even mean the ASF would have to stop using the name, much less give up their website, only that they would no longer be able to sue other people for using it as well.
If the Apache group of tribes claimed that the word "Apache" was offensive, and managed to convince the USPTO that that was the case, maybe they could get the trademark rescinded, but that wouldn't mean they would get it for themselves, nor would it even mean the ASF would have to stop using the name, much less give up their website, only that they would no longer be able to sue other people for using it as well.