Are you talking about Godel's ontological proof or Godel's Incompleteness Theorems? If the former, it is worth noting that Godel himself did not consider it evidence of the existence of any deity, let alone one of the Christian type. While it is an interesting exercise in Modal Logic, it only proves the existence of a god if the axioms are true. If you accept the axioms, then the Godlike being is certainly nothing like the Christian God.

The key axiom that makes Godel's ontological proof "prove" god is axiom #3 "The property of being God-like is positive". By definition, anything that is positive must exist- a property that does not exist cannot be positive or negative. So this axiom assumes that the property of being "god-like" is possible and since anything that is possible exists, then god-like properties must exist. You can replace "god-like" with absolutely anything you want, and the proof will logically prove that it exists. The recent news story is simply based on a couple of guys plugging the axioms into a computer and proving that the argument is logical. All it shows is that Godel's logical reasoning was sound, which isn't a surprise since he was a genius logician. It does nothing to prove that the axioms are true. Since the other axioms are completely contradictory to Christianity, it would be pretty easy to tear apart any Christian trying to adopt this argument. For example, within the argument a property is either positive or negative, it cannot be both. So the property of say inflicting pain must be either positive or negative, Christian theology adopts the view that pain is both a positive and a negative.

As far as Godel's Incompleteness Theorems, I have seen some half educated theists attempt that tact a few times over the years. The good news is that they picked up a math book, the bad news is that they don't understand it and usually remain willfully ignorant. The argument basically boils down to a fancy "God of the Gaps" argument with the theist refusing to understand that you can't cut and paste a theorem which requires a formal logic system as a condition and apply it to the universe in general. The universe is not a formal logic system as far as we know. They of course will throw out butchered quotes from Dr. Hawking (who is a genius that can't be questioned until you point out that he did not come to the conclusion that God exists, at which point he reverts to being an idiot atheist.) Dr. Solomon Feferman does a much better job describing the general public's misconceptions regarding the incompleteness theorems than I can. http://math.stanford.edu/~feferman/papers/Godel-IAS.pdf

I don't see why it would, having a computer do the calculation doesn't make flaws in Modal logic systems or the axioms disappear. The real importance of the experiment isn't that Goedel's argument is valid in a Modal system, it is that the computers are capable of quickly computing proofs using Modal logic which is an important step in eventually creating true artificial intelligence since Modal logic is more flexible than many other logic models in that it includes formalization of various qualifiers allowing for dealing with non-absolutes.