Plus, blanket = complete?
Context:
Critics of this idea will point out that the F.D.A. is equipped to review drugs from a health and safety point of view, not from a moral one. This is a reasonable point. But it doesn’t follow that the secretary or president should have a blanket veto over its decisions. Congress could easily establish a law providing for separate bioethical review.
(NYT article)
In this context, 'follow' means that the first statement does not necessarily entail the second statement. That is, 'The FDA may review drugs from a health and safety point of view, but that should not require that the secretary have a blanket veto.'
Here, the word 'blanket' is used to mean 'complete or absolute.' That is, '... the secretary or president should not have the ability to completely reject its decisions.'
"Follow" means logically follow. That the conclusion is forced by logic.
I say that when it rains, my dog gets wet and that when my dog gets wet, he stinks. Take those as true statements.
If I then tell you that it is raining, then it logically follows that my dog will stink.
In the text you quote, the author is saying that even if you accept as true all that is said, it does not logically follow that there should be a blanket veto power held by the president or secretary.