Yes. Middle-earth became Europe, the Shire became Great Britain and Ireland, and the One Ring was destroyed about 6,000 years ago. I will allow Tolkien to speak for himself. All quotes are from Tolkien's letters (unless otherwise noted):
‘Middle-earth’, by the way, is not a name of a never-never land without relation to the world we live in (like the Mercury of Eddison). It is just a use of Middle English middel-erde (or erthe), altered from Old English Middangeard: the name for the inhabited lands of Men ‘between the seas'. And though I have not attempted to relate the shape of the mountains and land-masses to what geologists may say or surmise about the nearer past, imaginatively this ‘history’ is supposed to take place in a period of the actual Old World of this planet.
May I say that all this is ‘mythical’, and not any kind of new religion or vision. As far as I know it is merely an imaginative invention, to express, in the only way I can, some of my (dim) apprehensions of the world. All I can say is that, if it were ‘history’, it would be difficult to fit the lands and events (or ‘cultures') into such evidence as we possess, archaeological or geological, concerning the nearer or remoter part of what is now called Europe; though the Shire, for instance, is expressly stated to have been in this region (I p. 12).6 I could have fitted things in with greater versimilitude, if the story had not become too far developed, before the question ever occurred to me. I doubt if there would have been much gain; and I hope the, evidently long but undefined, gap in time between the Fall of Barad-dûr and our Days is sufficient for ‘literary credibility’, even for readers acquainted with what is known or surmised of 'pre-history'.
I have, I suppose, constructed an imaginary time, but kept my feet on my own mother-earth for place. I prefer that to the contemporary mode of seeking remote globes in ‘space’. However curious, they are alien, and not lovable with the love of blood-kin. Middle-earth is (by the way & if such a note is necessary) not my own invention. It is a modernization or alteration (N[ew] E[nglish] D[ictionary] ‘a perversion’) of an old word for the inhabited world of Men, the oikoumenē: middle because thought of vaguely as set amidst the encircling Seas and (in the northern-imagination) between ice of the North and the fire of the South. O.English middan-geard, mediæval E. midden-erd, middle-erd. Many reviewers seem to assume that Middle-earth is another planet!
As for the shape of the world of the Third Age, I am afraid that was devised ‘dramatically’ rather than geologically, or paleontologically. I do sometimes wish that I had made some sort of agreement between the imaginations or theories of the geologists and my map a little more possible.
I imagine the gap [between TA and now] to be about 6000 years: that is we are now at the end of the Fifth Age, if the Ages were of about the same length as S.A. and T.A. But they have, I think, quickened; and I imagine we are actually at the end of the Sixth Age, or in the Seventh.
In (5) we meet the conception of the dragging of Tol Eressëa back eastwards across the Ocean to the geographical position of England - it becomes England (see I.26); that the part which was torn off by Ossë, the Isle of Íverin, is Ireland is explicitly stated in the Qenya dictionary. The promontory of Rôs is perhaps Brittany..
- The History of Eriol or Aelfwine1
So, as the other answers have already said, albeit with fewer citations, not only is Middle-earth in our own universe, it is on our own planet, and eventually became what we now know as Europe. The Shire became the British Isles (Great Britain and Ireland).
The LotR wiki produced this map showing how Middle-earth would probably align with modern day Europe:
1This was among the earliest writing Tolkien did regarding Middle-earth, and later, he decided against the idea of "dragging" an island to the current location of the British Isles; instead, he chose to say that the seas eventually rose and surrounded the Shire, turning it into the British Isles.