I'm not sure where we would begin, exactly -- and it sounds like what you're asking for is a beginning.  Nor am I well-enough acquainted with non-western traditions of thought to write about them with any authority.  But a few angles of approach to your question suggest themselves:

(1) First, doesn't the notion of synthesis dictate the answer you will get?  That is, I would want to remark that synthesis or resolution is not only a western notion but a specifically Hegelian one: it sounds like you're looking for a perspective from which both eastern and western philosophies become intelligible in a broader framework.  This positing of a more capacious perspective could often be useful when approaching the thought of others; but I would suggest that it may be precisely the entrance you have to avoid here.  In fact, Hegel himself gives such a synthetic account of "oriental philosophy" in his Lectures on the Philosophy of History and  Lectures on the History of Philosophy, and I doubt it will much help anybody who wants to approach eastern thought today.

What is wanted, I think, is some way of bringing eastern and western philosophical discourses in contact with each other.  Without the guarantee of a synthesis, this is very tricky; there have been lots of mediocre attempts to read eastern thought through a western lens, or to claim that such-and-such a philosopher sounds very much like some kind of eastern religious discourse.  Things are compounded by the unfavorable conditions under which we usually have access to south and east Asian traditions of thought (on which, more below).

2) One of the very first problems we're likely to encounter is the question of genre.  You say that both traditions share the word 'philosophy,' but of course this isn't really true; philosophy is a specifically Greek idea, and it's a good sign that, if we want to define what we mean by the western spiritual/philosophical mindset, as you say, then the Greeks (especially Platonism, but also Aristotelian logic) provide one of the two main streams -- the other being Christian monotheism.  And although these two streams have interacted, so that it makes sense to regard western ideas of knowledge as very much connected with ideas of God, there's also quite a clear separation between religious and philosophical discourses.  (So much so that religious thinkers have been known to deride the "philosophers' god.")  Approaching the eastern traditions -- and here I imagine you're thinking of the strains of thought we know generally as Hinduism, Buddhism, Confucianism, and Taoism -- westerners have always noted the difficulty of classifying them as either philosophy or religion.  Perhaps they have some concern with wisdom in common with western philosophy; but then, philosophy has seldom made wisdom a central concern, and very likely there are radically different notions of wisdom involved.

3) Your post already seems to acknowledge some of the difficulties: do we even know what western philosophy is, much less eastern philosophy?  Are these categories even useful?  As my crude sketch of western thought's two sources suggests, I think the notion of a western tradition is useful.  Your mention of scientific method suggests another approach to this tradition (centered on Descartes and the modern philosophy of the subject), and we could look to Heidegger and Derrida for more sophisticated analyses.  I imagine it could make sense to talk about eastern philosophy in a similar way, but I am less certain of how historically accurate this would be.  In China, for instance, three major schools (Buddhism, Confucianism, and Taoism) coexisted and interacted for centuries, giving historical grounding for some idea of Chinese thought.  But I'm not sure how much sense it would make to trace these traditions to a single origin.

The western tradition is also, of course, more complex than most university philosophy surveys allow.  People tend to avoid noticing shifts in a philosopher's thought; and this kind of attentiveness is needed in order to think about the meaning of the tradition without reducing it to some static body of propositions.

In regard to eastern philosophies, I think this problem of understanding detail together with an understanding of a whole tradition is intensified by our outsider's view.  This perspective gives us the impression, at one moment, of an entire philosophical tradition that is simple and internally harmonious and, at another moment, of a mystifying plenitude of writers whose place in a tradition is not so clear.

Sometimes it's not just an outsider's view that produces these effects; some interpretations, like "political Confucianism," have become so ingrained that they obscure the richness of Confucius' writings and replace these with received ideas.  (For this reason, it's sometimes even suggested that a westerner has an advantage in reading these texts for the first time.)

4) You mention "Western thought and Eastern non-thought" as a possible way of approaching the distinction.  I would sound a note of caution about these binary distinctions.  A problem is that they tend to inscribe the other--eastern thought--in a framework given by a certain logic that you already possess.  People will offer the claim that western philosophy is rooted in the individual while eastern philosophy is not, the contrast between action and passivity, the idea that western philosophy is a kind of secular monotheism while eastern philosophy is a quasi-religious paganism, and so forth.  These are all probably useful, to some extent ("somewhat true," as you've already said).  But, for instance, do we know what "passivity" is in, for instance a Buddhist context?  (Here, a comparison with a western philosopher could be useful: I would think that there is some sense in setting Buddhist thought about activity and passivity beside Emmanuel Levinas' radical passivity.  This might, at any rate, help us to see whether passivity in Buddhism is as simple as the active/passive distinction makes it sound.)

5) The questioning/listening image sounds a bit too good to be true.  It does suggest the primacy of knowledge in western philosophy and the image of a knowing subject who inquires after knowledge of the world, which was central to western thought for a few centuries.  But I'm much less certain whether listening characterizes eastern philosophy.  Maybe you're onto something.  But I doubt that western and eastern philosophies, grounded in their different histories, will fit together in this way like two parts of a whole.

6) Two remarks occurred to me as I read your post again:

- I certainly hope that capitalism isn't the end-all-be-all of western philosophy; at any rate, I doubt many philosophers would regard it as such.  (To return to Hegel, capitalism might be a kind of false absolute.)  Nonetheless, there is some justice in your statement.  It is also under the guise of world capitalism that western political philosophy currently encounters east Asian political thought, and the language of liberal democracy is taken up in the strangest places lately.

- I've already expressed skepticism about the notion of resolving eastern and western philosophy into a coherent discourse or way of life.  But in the long history of westerners infatuated with or interested in eastern thought and practices, you might find some interesting examples.  Ezra Pound comes to mind.  Pound was very taken with Confucianism of the basically "political" variety, and he published lots of translations of Chinese poetry.  He also published his friend Ernest Fenollosa's little book on Chinese characters, and this text alone makes clear that his understanding of Chinese writing may not have been exactly trustworthy.  (Fenollosa's and Pound's claims for Chinese characters remind me of Heidegger's interest in the pre-Socratics.)  But on the other hand, Pound may have regarded his orientalism as a sort of synthesis, and it was evidently productive for his work.

***

As you can see, I'm talking about the question, not trying to answer it.  I would be very glad to hear more direct responses from people better qualified to write about the eastern thought, especially since one of my interests at the moment is the (possible) dialogue between western political philosophy and Chinese political discourse.  It seems to me, though, that the one way to address oneself usefully to your question would be to attempt a reading of some central texts in eastern traditions.  Since your reading practices are probably informed by western modes of interpretation, I think this could be quite interesting.

It's a very dense topic and I'm certainly no expert. But I've always considered the two unresolvable. Though through literature, I think the two can meet and interact.

A lot of writers have explored the eastern tradition through their own western lens and through fiction. The two that come to mind are Hesse and Salinger. In Franny and Zooey Salinger in my opinion attempts to synthesize the religious aspects of western Judeo-Christianity and Eastern zen. Whether or not Franny succeeds in merging the two worlds is left up to the reader, but given her mental breakdown I think we can understand the magnitude of the clash.

As Jeremy says it is dangerous to reduce to binaries, but if we want to in the context of this book, we have a couple of options for representations of East and West. The Intellectual West vs. the eastern family. The characters in the book are stifled by both their keen intellects and their relationship with their family. At once they are endangered by both, but also protected. It is through her older brothers that Franny is led to her spiritual breakdown initially, but it is also through what they taught her that she can pull herself out.

Ultimately Salinger makes that oft-cliched conclusion that love is the only significant world order. And maybe it is cliched because it is true. Maybe the spirituality of the east and the west meet with love for humanity.

In any event, whether the characters synthesize their intellect to their spirituality is not as important next to Salinger's exploration of the matter through fiction. That he is applying his world to the far east, in my opinion, is the successful synthesis of the two worlds. He makes no universal conclusions, only a personal response. He resolves the tear between his own individuality with that of his audience, he resolves his intellect to humanity.

On a somewhat sidenote, the main plot device of the book is Franny's use of The Jesus Prayer which emerged out of the Russian orthodox practice. And isn't Russia the symbolic conjoiner of east to west? 

Wow!  I confess that I am overwhelmed by the breadth of the discussion.

I am very familiar with Buddhist thought but not with other Eastern Philosophies or Western philosophy in any depth or breadth.  My familiarity with Buddhism tends to confirm the expressed sense of un-groundedness and speculative nature of much of the above.  I, like the others, am unclear what Mark means by “synthesis.”  I tend to think it is along the lines of: The combination of ideas into a complex whole – or something close to that. 

In other posts I have suggested that science is the exploration of the objective and religion the exploration of the subjective; maybe philosophy is more in the religious camp.  Although, apparently, quantum mechanics is pushing the scientist, or at least physicist, closer to the philosophers.  I believe more than one physicist has pointed out that; there can be no complete explanation of how the world works that does not include consciousness.  It is unlikely that consciousness will be reduced to an objective definition – or at least any more objective than light or gravity or the strong force etc..  Maybe there is no objective reality?

The search for synthesis of philosophy seems, to me, like trying to synthesize all of the wild flowers in the field; each of the flowers is complete and true to itself and the cacophony of them all is rich beyond measure.  Maybe that is equally true of philosophy, religion and science; they are each rich in their own right and the chaos of the aggregate is immeasurably entertaining, if nothing else – – – – as long as we don’t think that we actually know the truth or can know the truth.