The problem with using Hawking as an example of what supposedly might have happened under a non-Judeo-Christian regime, is that Hawking was perfectly healthy at birth, and continued to be so well into adulthood. Nobody would have wanted to euthanize him.
He is an exceptional man, not only for his brainpower, but because he has lived with ALS (Lou Gehrig’s disease) for decades longer than about 99% of other sufferers.
The author also makes the typical assumption that only alternative to a Christian society is to live in some sort of atheistic dictatorship. Several countries in Western Europe can barely be called Christian these days, including Scandinavia and the UK. In fact, most of those societies have been overwhelmingly secular for decades now. Yet somehow democracy continues to thrive in those places despite all the scare stories you hear from the Christian fundamentalists in the US.
Once a society rejects that foundation and applies to individual rationality, that is when the risk of misstep is at its greatest.
The cultures and government of Sweden and the like, the rationalists now retreating feverously from democratic socialism, are not great banners to hold aloft, IMHO.
I do agree that there are weaknesses in the arguments made by the author, they are not however sufficient for me to pitch a couple of millenium of western civilization aside and let Hitch be my guiding light.