There are several interviews in Quine in Dialogue that are worth reading. A 1978 interview with Magee has a bit that made me stop.

Quine, in a discussion of modality, says, “There is a fashionable philosophy of possible worlds, but it is something undreamed of in my philosophy.” Rhetorically, this is kind of cute. What interested me about this bit is that it is clearly after Kripke’s work on modal logic. I don’t remember Quine saying anything more explicit about Kripke’s work or the subsequent development of modal logic, so this oblique remark is the most I can remember Quine saying on this matter. (It is likely I’ve overlooked something in his corpus.) 

This got me thinking that it is a little surprising that Quine never responded to the technical developments of modal logic that occurred in his lifetime, since they would seem to impinge on his claim that alethic modality is an obscure notion. A possible reason for the lack of response, if there is a lack, is that Quine did not see the technical work as engaging with his complaints, or not engaging enough. Are there any articles out there that lay out how the development of Kripke frame semantics for modal logic respond to Quine’s criticisms of alethic modality? Surely there must be. I’d like to read some such paper since it isn’t clear to me at the moment how the argument would run.  

Quine’s lingering doubts about modality, it seems, would have to be based on the epistemology of modal knowledge, since the semantic issues would be settled by Kripke frames. Roughly, it seems like there could be two sources for his doubt, his broadly empiricist views and his broadly naturalist views. (This isn’t going to be terribly helpful at pinning things down since it would require more specification  of Quinean empiricism and naturalism than I’ll be giving.) There isn’t much room in his view of epistemology for modal knowledge. Other possible worlds do not causally impinge upon us. Also, since his characterization of translation is extensional, it puts up something of a barrier to intensional operators like modal ones. That being said, I suspect that it his naturalistic views that are the source of the doubts. (Again, this requires a sharp characterization of naturalism for it to have any force. I’m having some difficulty getting it into focus.) Roughly, it seems like Quine doesn’t think that science requires the full-blooded sort of possibly worlds that one finds in Kripke or Lewis. Rather, when Quine talks about possibility, e.g. possible utterances in the context of grammar construction, it is cashed out in terms of the output of a Turing machine or context-free grammar. Possibility in different areas of, say, physics would likely be cashed out in terms of Hilbert spaces or whatever appropriate mathematical space is used. I’m not sure exactly how this fits with his comments on dispositional terms, which I’m not that familiar with. Quine’s doubts about modality could be the product of a combination of these things though.