This belief is fine as far as it goes. But what ends up happening with secularists like Carroll is that it is only negative conclusions about religion and the supernatural that are admitted into the scientific discussion; positive conclusions need not apply. Not only that, but, using their methodology, negative conclusions are assured. Just go back to Carroll's recent discussion of the immortality of the soul:
If you claim that some form of soul persists beyond death, what particles is that soul made of? What forces are holding it together? How does it interact with ordinary matter?In other words, any supernatural entity must comply with exclusively natural criteria, in which case, well, it wouldn't be supernatural any more, would it? So, under Carroll's method of verifying supernatural entities, a negative outcome is the only outcome possible. The game is fixed from the beginning.
This also means, ironically, that naturalism, under this methodology, is unfalsifiable--the favorite demarcation criterion of secularists between science and non science. In other words, under their own definition, this kind of naturalism is not science, since it is not falsifiable.
It's a crazy world.