How hard would it be to create a protein "by chance"?
In the original column for The Claremont Review of Books, David Gelertner does not suggest starting from atoms, but from amino acids. This changes the calculation slightly:


  The total count of possible 150-link chains, where each link is chosen separately from 20 amino acids, is 20^150. In other words, many. 20^150 roughly equals 10^195, and there are only 10^80 atoms in the universe.


In any case, these kinds of estimates are purely theoretical and are not hugely useful. We do not know how early proteins evolved, but they did not have to just assemble from scratch.



As mentioned in the comments, this kind of argument for creationism (or 'Intelligent Design') is like a card trick - you start by dazzling with vast numbers, and use them to conceal the logical sleight of hand.

The premise of his argument is simple - there are a vast number of possible sequences, only a small fraction of those can fold into functional proteins, and there have not been enough mutations to make up the difference.

There is this strange argument: 


  But what does generating new forms of life entail? Many biologists agree that generating a new shape of protein is the essence of it


I'm not sure who these 'many biologists' are, but the relationship between novel folds and new species is not clear to me, at least.

Anyway, the second part of his argument relies on the work by Douglas Axe on estimating the fraction of functional folds (see this paper for example). Although he describes him a bit oddly ("Axe is a distinguished biologist with five-star breeding") it's reasonable to make this kind of estimate.


  He estimated that, of all 150-link amino acid sequences, 1 in 10^74 will be capable of folding into a stable protein. To say that your chances are 1 in 10^74 is no different, in practice, from saying that they are zero. It’s not surprising that your chances of hitting a stable protein that performs some useful function, and might therefore play a part in evolution, are even smaller. Axe puts them at 1 in 10^77.


So here is the very small number to contrast with the very large one of the first step. Now for the 'bridge' - the large number of attempts that would be necessary to make a very rare event (a folded, functional protein) possible.


  Suppose, then, that every bacterium that has ever lived contributes one mutation before its demise to the history of life. This is a generous assumption; most bacteria pass on their genetic information unchanged, unmutated. Mutations are the exception. In any case, there have evidently been, in the whole history of life, around 10^40 bacteria—yielding around 1040 mutations under Axe’s assumptions.


So the argument - roughly - is that 10^-77 * 10^40 is still very small. That is, mutations are not frequent enough to overcome the extreme rarity of functional sequences.

The sleight of hand, I think, is to confuse abiogenesis and mutation. When talking about the entire 'landscape' of possible sequences we are talking about abiogenesis - randomly picking 150 amino acids (say) from an alphabet of 20. When talking about bacteria we are obviously talking about mutation.

We really have no idea how the first primordial proteins formed, or what their properties were. They could have used a smaller set of amino acids (although this does not change the numbers much), or could have assembled from multiple smaller peptides, or even been partially folded.

On the other hand, mutation of existing proteins necessarily happens on sequences that can already fold. Moving to a sequence that is a neighbour in fold space is very different to picking a completely new point in that space.

So really, the numbers do not tell us anything because they relate to separate questions.