This is horrific, but not much different from partial birth abortion. If one takes a purely logical and not spiritual and religious view of life a lot of people will need to die as a matter of convenience and efficiency and it does not take a Hitler to start the process.
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.