Can an article published in a non-peer reviewed venue later undergo peer review?
I haven't heard of this happening, but I suspect what would happen in your example is that the author would write an "extended" version of the article, perhaps including a few additional results or more details.  They could submit it to Science or wherever, which would review it like any other paper.  It's not duplicate publication because it's a different paper, and Science would likely still consider it "novel" because it hasn't been published in an academic journal.

They should certainly inform Science of the previous publication, but I don't think it'd be a barrier to republication if the paper really was good, and if the additional content was substantial.

In practice, it'd be necessary to rewrite the paper anyway, because the audiences and the expectations of the outlets would be completely different.  The New York Times wouldn't want a paper full of technical details, and Science would insist on having the technical details.

If there were copyright issues (e.g. the Times held the copyright and wouldn't give permission to publish an adapted version in Science), then the author could rewrite the text of the paper.  This makes it a different work for copyright purposes even if they both describe the same results.

I've never heard of any system of "certifying" a previously published article as peer reviewed, like you suggest.  However, there is the somewhat related concept of an overlay journal, in which the journal effectively "certifies" a preprint.  The journal provides the peer review, while arXiv provides the hosting.