That is not true. Just as Congress could grant citizenship to all Native American tribespeople born in the United States by simply passing a law, Congress can also remove birthright citizenship by simply passing a similar law.
Indians are a different case, and a limited one. They were treated as having sovereign status even though they did not have sovereign territories, i.e. they were here first, so they get special treatment.
The law you cite just put them on the same basis as everyone else, gave them the same rights that everyone else born here had, it didn't take anything away from them.
If you want to take anything away from people, you're going to need a Constitutional amendment.