To clear things up, just in case you were referring to me in regards to it being my relatives, I don't believe in this ideology (yet, at least) and I don't plan to.
With regard to that information about the Greeks, Romans, and Vikings; that is interesting and seems to only reinforce the idea that civilizations that advance the human position on Earth can obey these "laws of the jungle".
"If it is your relatives you might think differently."
I wouldn't doubt it that some would attempt to save their own family from such oppressive behaviors, but at the same time, they might be considered for execution themselves, for costing society that much more in resources by resisting what might be viewed as a "practical" ideology.
"Those with disabilities are not likely to mate so there is no cause for alarm regarding species degenerating."
I empirically agree, and it is a good point for not killing people for that reason alone.
"Look at Stephen Hawkings the foremost physicist of the day. By this Nazi like mindset, he would be set for the furnace"
This is true, and I've brought up this argument before. Let me present you with this person's response:
Stephen Hawking has already pulled more than enough weight to constitute him being given the right to live. Though, even if he was killed beforehand, someone else, who wasn't disabled, probably would have concocted the theories that he did eventually.
EDIT: I forgot to mention, that disabled people could be justifiably killed under this ideology just for wasting resources, no matter if they reproduce or not.