Sir Nicholas was absolutely correct to answer the question openly when asked. As others here have rightly stated his response was a statement of fact and he has also made it clear that it was not just a matter of opinion. Corbyn set out his position on the use of nuclear weapons publicly and that has the potential to completely undermine a cornerstone of Britain's defence, something that CDS is duty bound to uphold and to make sure the public understand. That is not politics, it is a competent leader performing his duty.
If Corbyn feels it is a personal attack it can only be because someone far better qualified than he to speak on this issue has demonstrated why the public statement of his personal ideology is dangerous to the security of the U.K. CDS has not criticised the man, just the public announcement of his position and the undermining of UK Defence Policy in (IMO) a very foolish way. I should add that position is not even Labour policy.
It's Corbyn that has made himself look like a leftie fool, not CDS and he is understandably feeling pretty stupid about it - hence his outburst.
Well, spoken, Sir Nicholas.