In his notoriously laconic reply to Stroud, Quine treats Stroud's dilemma as merely an instantiation of the sceptical worry that reality might be very different from how we perceive it to be. If the first horn of the dilemma, that science is wrong, obtains then it naturally follows that reality is not as we perceive it to be. If the second horn of the dilemma, that all our theories are merely unverifiable posits, obtains then there is no way for us to rule out the possibility that reality is not as we perceive it to be. Both horns of Stroud's dilemma, then, amount to the sceptical worry. I take this conclusion to be uncontroversial, and I will therefore treat a successful reply to the sceptical worry as a successful reply to Stroud's dilemma.
Quine maintains that the meaning of words is determined by their usage within a linguistic community. When we say: "there is a spoon," we are simply referring to a certain set of sensory stimulation’s to which our linguistic community has inter-subjectively applied the label "spoon." Ontological worries regarding the nature of what lies behind sensory stimulation is irrelevant to the relationship between language and these sensory stimulations. In short, what science regards as a "real" spoon is just that inter-subjective set of sensory stimuli:
What then does our overall scientific theory really claim regarding the world? Only that it is somehow so structured as to assure the sequences of stimulations that our theory gives us to expect. More concrete demands are indifferent to our scientific theory itself. (Quine , p. 474).
This does not of course mean that our scientific theories can never be mistaken. However, they will be mistaken precisely in terms of a failure to accurately predict observation. If, however, a scientific theory were to account for all possible observation, in what sense could that theory be described as mistaken? Once this question is viewed within a naturalistic framework, Quine insists that the answer is none. The sceptical worry that reality may be radically different from what we take it to be rests on the assumption that there is some description of objects prior to the scientific description by which the scientific description may be evaluated. However, once we have located epistemology within science such worries become obsolete. If sceptical possibilities do not affect science's ability to predict observation then they are irrelevant. If, on the other hand, a sceptical worry does impact on science's predictive power then that worry does fall under the umbrella of science and its dissolution becomes the task of science.
For a detailed and extended discussion of Quinean naturalised epistemology see Ben Bayer's paper, Varieties of Naturalized Epistemology: Criticism and Alternatives.
Quine, W.V.O., "Reply to Stroud." Midwest Studies in Philosophy vol. VI The Foundations ofAnalytic Philosophy. Edited by Peter A. French et al. Minneapolis MN: Minnesota Press,1981.