There's a good case, IMHO, for "unreasonable doubt" but as a practical matter it can usually be ignored. Whether a given religious text is the word of God is a separate debate from whether God exists, however. The latter almost inevitably comes down to a matter of faith even when the text in question is internally consistent and makes rigorously logical arguments, but the latter can be evaluated in rational terms without resort to a text arbitrarily chosen as authoritative.

Let's consider the universal singularity subjected to the Big Bang, since its only detractors are religious fundamentalists who obviously aren't atheists. It's static matter; it has no internal motion (since it's a 0 dimensional singularity there's no space in which anything can move) nor external motion, and can acquire none from any source because it contains the entire universe, by definition. Anything with kinetic energy that MIGHT be present must simply be absorbed by the singularity, its energy of motion transformed into additional matter, per the Second Law of Thermodynamics. The First Law of Thermodynamics precludes spontaneous generation of a kinetic energy source to put the universal singularity in motion. Consequently, the Big Bang cannot be the result of a natural cause and therefore must be the result of a supernatural one; when you have eliminated the impossible, whatever remains, however improbable, must be the truth.

We don't have to call that supernatural First Cause "God" if that makes people uncomfortable; we can simply call it "a supernatural force responsible for setting in motion all of existence." However, I think any reasonable person would agree that's what's generally understood by "God" whether we're willing to admit it or not. Even if we posited some as yet unknown exception to the Laws of Thermodynamics, it wouldn't be because of supporting observational evidence, but solely because we irrationally dislike what all observational evidence DOES support. It would be positing, on the basis of faith alone and in defiance of all known facts, a violation of established natural law, and thus still something supernatural; indeed, that it's supernatural is about the only thing we COULD say about it with certainty. So our rational evaluation of the evidence leads inescapably to one of two places:

1) The conclusion beyond reasonable doubt that something fitting the definition of a deity exists or

2) Outright rejection of fundamental physical laws.

Unless we ignore the principles that matter and energy cannot be created or destroyed (the First Law of Thermodynamics) and/or that entropy within a closed system always increases (the Second Law of Thermodynamics), existence itself defies a purely natural explanation. We can get a pass on the idea that an object at rest (like the singularity that preceded the Big Bang) tends to remain at rest unless acted on by an outside force (Newtons First Law of Motion) only because Relativity has demonstrated exceptions to the laws of Classical Mechanics, but Thermodynamics is still inexorable. No one else claiming to have discovered a perpetual motion machine is taken seriously, so I fail to see why those who claim the WHOLE UNIVERSE is one merit an exception.

Denying the first two laws of thermodynamics or Jovian moons for the sake of a cherished belief are both dogmatically irrational. When that's compounded by claims to have proven a negative we're far into the realm of fantasy. While everyone is entitled to whatever fantasy they wish to indulge, that doesn't make it credible, and no one is entitled to inflict their fantasies on others. Deism vs. theism seems to me a far more plausible and constructive debate than atheism vs. either; neither deism nor theism inherently imply a particular religion, but both naturally (and, IMHO, accurately) SOME religion.

Agnosticism has ample room for both, a typical example of why I find agnosticism far more reasonable than atheism, as well as the reason I suspect the term "positive atheism" was created: It's a great way to coopt reasonable positions like skeptical and "non-denominational" agnosticism while maintaining a repudiation of them. If all agnostics are treated as a subset of atheism, atheism as a whole instantly becomes far more reasonable, and religious belief as a whole far less so. Unfortunately, that's a transparently cynical and grossly inaccurate representation. From where I sit, there are three general groups of people:

Religious People
People who recognize and admit they accept, to varying degrees, a given religion, usually informed, again to vary degrees, by observation and reason.

Agnostic People
People undecided about the existence, function and/or identity of a deity or deities.

Atheist People
People whose rejection of any and all deities is as much an article of faith as affirmation of one or more is for the first group.

If you want irrational, compartmentalized, cognitive dissonance, that last group is probably the best place to look, because atheists invariably insist their core religious beliefs are an unavoidable consequence of core scientific beliefs, despite fundamental and insoluble contradictions between the two. The principal difference between them and (other) religious fundamentalists is not denying what others affirm, but that they neither recognize nor admit basing their position at least as much on belief as the first group does.