[Lush]

...And let's not forget Lúthien, who is named the greatest of the Eldar, and who is, perhaps, more courageous than all the sons of Fëanor combined.
I think that the problem, if there is any, is not that Tolkien purposefully excluded women, it's just that he had difficulty developing believable female characters. Galadriel, Arwen, Goldberry, Lúthien, and Melian are fairy-tale creatures that, though presented as wise and noble, are nevertheless placed on a pedestal so high that we cannot examine them up close. They are mysterious and enchanting, but they are also remote, inscrutable. We don't get to read about the things that go on in their heads, and we certainly don't get very much of a glimpse into their everyday lives as we do with the male characters. Why? Because writing about women's inner lives wasn't something that Tolkien could do well. He was no Chekhov in that regard.
Éowyn is the only exception to general rule, here we have some emotional struggles, suicidal tendencies, anger, boredom, lust, fear, etc. It is as if this shieldmaiden was Tolkien's only real stab at creating a woman with some texture and depth; and her brief presence strikes a nice balance between the major (if you can even call them 'major') female characters in the LotR. We have Arwen, the distant beauty, Galadriel, the wise and powerful leader, and Éowyn, the real human being, which, I think, gives the book enough oomph. Even a snobbish, picky reader such as myself (I am forver whining for more characters to "indetify with") is kept interested and awed throughout the story. On my first reading of the LotR, I was delighted to "discover" Éowyn among furry-footed Hobbit comrades and sword-clanging leading men, mainly because she was so unexpected, especially when I recalled the "ornamental" Arwen at the beginning. Now I understand that Arwen is part of the mystery, the enchantement that is the LotR, and I wouldn't have her any other way (I don't, however, mind PJ's changes), though a bit more information on her reign as Queen would have been nice.
Overall, do I think that Tolkien was prejudiced? Do I think he had a beef against women? No. I just think that he stuck mostly to what he knew and understood. His strengths as a writer did not include real women, this is a fairy-tale after all, not Lifetime (thank God), and not Chekhov. If I want a character to "indetify with", I can go for Erica Jong.