No, English 'fiddle' isn't derived from Gaelic, though both languages (and most other European languages, come to that) have similar words meaning much the same thing. 'Violin' (a late coinage) is one of them.

Ultimately, though, the etymologies of words relating to musical instruments, and the origins (real, putative or imagined) of those instruments are completely irrelevant when it comes to the business of 'national' instruments which, like other 'national' emblems, are merely symbolic. Countries like Scotland, Wales, Finland and so on had a big stake in the Romantic Nationalist movements of the 18th and 19th centuries, and felt a great need for such symbols to assert their individualities in the face of the growing power and influence of larger neighbours.

Demonstrating at great length that such-and-such an instrument isn't originally English is pointless, as virtually none of the 'national' instruments originated in the countries that adopted them as emblems, though some were refined there into the forms we know today. The only European 'national instrument' I can think of that is actually played by a significant proportion of the population of the country concerned is the Finnish Kantele, which is taught in schools. Doubtless there are others, but the point is that it really doesn't matter.

The whole concept of 'national identity' is largely an artificial construct; a useful tool through which powerful people manipulate the masses. Ultimately all such things are an extension of basic tribal instinct, which seems to be built into humanity (and probably all social species) at a genetic level; and which is in itself neither good nor bad, though it can be the agent of both.

As has already been remarked, it's often the case that what we might call 'culturally displaced' people seek to compensate for their lack of tangible roots by imagining an historic identity as a model to aspire to. Hence all those 'New Age Celts' with their fantasies of woad, shamanistic ritual and ancient pre-Christian bodhrans; and the 'Merrie England' folk, too; all madrigals, codpieces and recorders (I'm baffled by the pottage, though). You'd think that anybody who actually knew anything about folk music would have seen through all that long ago, but unfortunately the fantasists have always had a strong influence; and for as long as people think they can become well-informed by reading a couple of out-of-date books and a few wikipedia articles, we will continue to suffer all manner of nonsense.

A national instrument for England? Too late for that by a couple of centuries, I think; but since such things are symbolic, they need only to evoke a suitably 'national' image for a large enough number of people. That being the case, bells or brass would serve well enough; and it wouldn't matter if they were invented on Mars.