Apparently not. I started off in the stacks. Hofmann's Etymologisches Wörterbuch des griechischen produced the following:
νᾱός, ion. νηός, att. νεώς: m. Tempel, Heiligtum. Als Götterwohnung aus *νασ-ϝ-ος zu ναίω (aor. νάσ-σαι) wohne.Under ναῦς (ion. νηῦς, gen. νη(ϝ)ός or νεώς) there is no useful information, except a comparison to Indogermanic *naus, and Latin navis (war mir schon klar), so the root seems to be pretty basic, and, more importantly, entirely unrelated to the word for temple.
This still leaves the question, why 'nave'? The OED suggests it is derived from a post-classical use of the Latin navis or an Italian or Spanish equivalent. Not much help so far.
A little more searching on the web turned up an article from 1926 which makes some doubtful assumptions (etymological and otherwise):
The ship was for the Greeks something sacred and was regarded as the gift of the Olympian Gods. The economic life of the population around the Aegean Sea was based on shipping. It was no mere accident that the inner shrine of Greek temples was given the same name as the ship, naos. The word nave used in the Christian churches to this day is probably derived from the Latin word navis and handed down from the days of the temples, and it is known that in the early Christian Church the ship stood as a sacred symbol.(Hovgaard, William. "The Arsenal in Piraeus and the Ancient Building Rules," Isis 8.1 (1926): p. 18.)
Another article provides more substantial evidence for the Christian connection between the church and ships. The etymological claims are also much more conservative:
NAVE (Gr. ναός, εὐκτήριον τοῦ λαοῦ; Lat. navis, capsum; Fr. nof; Ital. Nave; Ger. Schiff, Langhaus) Authorities are not agreed on the etymology of the word, some deriving it from (1) ναός, temple, which is the ordinary Greek term for what we should call "the body of a church;" and others from (2) navis, a ship. The fact that in several European languages (e.g. French and Italian), the corresponding word is used to designate both "ship," and "part of a church" may be thought to favour the latter hypothesis.[...]As being the receptacle of the people, it was no great stretch of fancy to speak of it under the figure of a ship. The Ark was at all times the Old Testament figure of the Church. The idea of the comparison between the church and a ship was elaborated very early.[...]The resemblance of nave to its Greek equivalent (ναός) may be nothing more than accidental.(Smith, William, and Daniel Cheetham. A Dictionary of Christian Antiquities, v. 2 (1880): p. 1381)
So it seems we can trace this usage to the early Christian period, although it still does not explain how the Latin word navis came to be applied to architecture. Was this inspired by a perceived resemblance between the shape of a ship and the shape of the main chamber of a temple or church? Or is it based on a false etymology of the Greek word naos? I would tend to speculate the latter. Certainly the Greek words for 'temple' and 'ship' are similar enough to be easily confused, and this sort of etymological play was not untypical of the Romans -- or the Greeks, either, for that matter. However, this seems to be a post-Hellenic usage, not a connection that the Greeks themselves drew, which was my original question.
A final note is in order, however: In the process of this research I came across an interesting article (for those with academic journal access, here) on analogies between temples and ships in Homer. He notes the use of νηοὺς - - ˘ θεῶν ("temples of the gods") in Od. 6.10 and suggests word play with a common formula for "swift ship", for example, νηῦς - - ˘ θοή in the same metrical position in Od. 24.299. He cites a number of examples of possible cultural associations of temples with ships; I did not find them completely convincing, but I'll cite his explanation of the etymology here.
The nominative singular of νηός, “temple,” is identical to the genitive singular of νηῦς, “ship.” One often finds such superficial similarities in Greek and elsewhere between forms originally quite distinct: νηός “temple,” derives from proto-Greek *naswos, the verbal reflex of which is *nas-io > ναίω, v, “to dwell,” a fact not lost on the author of the Homeric Hymn to Apollo (cf. νηὸν ἔνασσον, 298), while νηός, “of a ship,” comes from proto-Greek *nawos and ultimately from PIE *neH2ues,which is widely attested. (544)He also makes a couple of intriguing remarks on connections between power/leadership and seafaring - this metaphor makes sense for a culture where seafaring is of supreme importance - as well as the connection of ships with temples in another context: their presence as dedications. (For the sake of readability I've eliminated his notes on sources in classical writers here; those interested are referred to the original article.)
The equation of temple and ship gives special point to the fact that as helmsman of Olympus Zeus 'sit[s] high on the benches' what the ancients rightly saw as a seafaring metaphor....Ships and parts thereof made common dedications, like the Roman rostra, usually at temples, and Christian lore says that Jesus, who like the Egyptian ferryman of the dead, Mahaf, was a fisher of souls taught from the boat of the Rock (Simon Peter) on whom he would build his church. (544)(from Griffith, R. Drew. "Temple as Ship in Odyssey 6.10," American Journal of Philology 123 (2002): 541-547.)