Not modern organic chemistry, but the original meaning of the phrase. Living things include extremely complicated molecules which are hard to synthesize. The original hypothesis called "organic" was that this is impossible. The claim was that atoms and molecules followed different laws of motion if the were parts of living things. This hypothesis was holistic -- the idea was that particles can't be understood unless you know of what they are parts. The alternative is that particles obey universal laws so you can understand what they do knowing only which universe they are in. Another way of putting it is that the original organic chemists believed in a force different from gravity and electromagnetism which only pushed things inside living things around. They called it, well mostly some German word which is translated as "vital force".
As an aside, many passages from long ago which we tend to read as metaphors were meant more literally than it we tend to imagine ( that sentence is. So hedged that it tends to be meaningless).
Then this German organic chemist synthesized urea and was pissed. Big deal you say, since you do that every day, but he synthsized it in a test tube. I guess it tooK a while, but this convinced chemists that the original organic chemistry hypothesis was all a big misstake. It is now mostly forgotten, but a view which dominated a field for a while is really genuinely gone, discarded on the ash heap of intellectual history.
The word organic with its holistic conotations survived in the social sciences.
Disclaimer:I really had no idea that I was going to write the last sentence of this post when I wrote the first.
I mention old organic chemistry just to note that the tree of knowledge has, or has had, branches which bear no fruit (that one has been pruned). Regular readers can guess that I am thinking of contemporary economic theory, and in particular, about the effort in the past 39 years to provide microfoundations for macroeconomics. An analogy is clear -- the claim that this simply can't be done is like old organic chemistry. It is the claim that whole economies can only be understood as wholes. This is not the analogy I have in mind. No one claims that macro can't ever be micro founded. The harshest critics of the effort to microfound macro think that the problem is partly shortcuts to easy but false links from micro to macro (like the representative consumer) but mostly that standard micro theory is psychology and in particular very bad psychology based on assertions about individual behavior which have been tested and rejected by the data.
No the analogy I have i mind is that the 39 year long effort might be a fruitless branch like organic chemistry. The difference is that there seems to be no empirical failure (like urea) which causes the adherants of the school to question their basic core hypotheses. So I think that recent research in macro theory (definitely including my own current research projects) isn't worth piss.