Yes he explicitly describes many of them as white. Though, as the other answers say, he also has Southrons and Easterners though they play only a small part in the LOR.
I think it is for the same reason that the characters in most Japanese stories are all Japanese, in African stories all African and in Indian stories all Indian - that the author came from Europe and he borrows from European myths and legends and stories - and was a specialist in Early English (that was his field as professor at Oxford University), and related European languages- and cast his tales in an equivalent of Europe.
MOST OF THE SPECIES, NOT JUST HUMANS, DESCRIBED AS BASICALLY CAUCASIAN IN APPEARANCE ORIGINALLY
Not just the humans, and the elves, and the wizards, and the dwarves, and the hobbits (I mean apart from the obvious differences from humans but basically smaller or more graceful etc versions of caucasian), also I'd say arguably the trolls and orcs are basically caucasian also.
Note that the bad guys as well as the good guys are caucasian too- Sauron was a student of Melkor, not sure if he mentions his colour, but artists generally show both as white.
The black riders were only black because they wore black cloaks and rode black horses and were invisible. They were of similar lineage to Aragorn, kings, also sorcerers and warriors, that were given the rings in ancient times and were soon bent to the power of evil. They were almost certainly all white also.
And Sarumen also of course was white.
I didn't realize it, but following up one of the other answers here, apparently the dwarfs are inspired by Jews - and their language inspired the Jewish language. Are Tolkien’s dwarves an allegory for the Jews?
TROLLS
The trolls, not described as dark skinned, just tough and strong. Often shown white in artist's illustrations.
It is hard to tell from Tolkien's own illustrations:
Also he doesn't say in his description, just says they don't feel anything if you try to hurt them, and that kicking them is like kicking a stone.
But since he drew inspiration from the Scandinavian trolls, surely he had in mind white trolls also.
Treebeard says that the trolls were made by Melkor in imitation of Ents. Though his testimony here is potentially fallible, in the world of the story, see below.
ORCS
The orcs are described as sallow skinned - i.e. an unhealthy pale or yellow colour.
Tolkien says the Ainur believed Melkor bred them from elves originally - Treebeard says they were an imitation of elves. Since Tolkein says none of his characters is omniscient, then we can't say which of them (if either) was right.
Either way - seems their natural healthy colour, if they lived on the surface like humans, would probably be white.
(By white here I mean, or course, that mix of reds, pink, light brown, shades of orange, and various other pale blueish and greenish and yellowish tones that for some reason we choose to call white - if you are "white" try putting your hand next to a sheet of white paper - is it really the same colour :). ).
They do also have slant eyes. So - that might suggest oriental in appearance when combined with sallow and flat noses. Apart from Down syndrome of course, but don't think he intended that.
But the rest of the description of orcs doesn't match oriental people at all - bow legged, long arms, and wide mouths, with fangs :).
So - I think- in the "in world" mythology at least - the Orcs also might well be best thought of as basically caucasian in origin, in the general sense - like the trolls.
That of course covers a wide spectrum of skin tones - and the slant eyes and flat noses best thought of similarly to the wide mouths and long arms, bow legs and fangs, as a result of the breeding experiments of Sauron and Melkor.
Not sure how it works biologically, but seems that, for whatever the reason, we all have genes that can express themselves as epicanthal folds and flat noses.
So, surely it's not that hard for those characteristics to arise, either selected for or as a side effect of other changes, in forced breeding experiments.
THEME OF INTERSPECIES CO-OPERATION, OVERCOMING PREJUDICES AND MUTUAL APPRECIATION
His books have as their main theme people working together and appreciating each others capabilities and overcoming prejudices, not just of different races but also of different species.
So in all this when I say "caucasian" - it's just a similarity of features and skin tone, not actually shared origin.
So - it is true that humans and elves could breed together, so not totally different species in the biological sense, though elves became mortal as a result of breeding with a human, some sort of magical thing there, that if an elf takes a human as a mate, then he or she becomes a mortal - BTW by "immortal" he only means very long lived, as long as habitability of the Earth, not truly immortal, see Lord of the Rings Interview with Tolkein
Also humans and elves were forcefully interbred with orcs possibly.
But apart from that, I think they are all not just different in ethnicity - but different species, which might not have been able to interbreed at all.
No examples of hybrids of dwarves with any others that I know of, for that matter, no examples of hybrids of hobbits with any other species, or ents ditto - and no interbreeding of elves with anyone else except humans.
CONCLUSION
But as for most of his characters being Caucasian in general appearance, I see that as no more surprising than most of the characters in Japanese, Indian, African, Chinese, etc folk stories, and also many more modern stories, also movies, being of their corresponding ethnicity.
It would be reasonable for a film to have some Southrons and Easterners in the more cosmopolitan places.
In the movies, then - I understand the limitations of needing to use human actors, but to my eyes, the various non human species are too similar to humans to be really credible. Probably you would need cgi for them all to give a good visual correspondence to the stories.
I have many other issues with the movies, as a Tolkein fan, seems he left all the best bits out :). E.g. Tom Bombadil. I can understand how you might think he is not essential to the plot - but he is the one character in the entire book that the ring has no effect on. Nor is he interested in the ring except as a trinket. Gandalf says he would be a most unsafe guardian as he would lose interest in it and most likely lose it. That extra perspective I think makes a difference to the entire story. Indeed, as a Tolkein fan, I'd go so far as to say that without Tom Bombadil it's really not the same story at all.
Plus doesn't have quite the same gentle humour. And in the stories, just about all the characters burst into song at the slightest opportunity, even the barrow wight. I think that hearkens back to C19 England where according to contemporary accounts people would sing as they worked - a tradition still common in some rural societies world wide but sadly lost in most modern "civilized" places. I think the movies should have done the same. Actually I skipped through parts of some of the films - some parts were really good e.g. visual depiction of hobbiton looks pretty much like I imagine it. But some of the action sequences he interpolated into the story seemed pointless and beyond credible, and went on for ages.
(just trimmed out a bit about whether Tolkein showed unintended racism, will put that into comment as realize the question wasn't about that).