Women who take up arms to fight are rare in Tolkien's universe, the only real example being Eowyn, who disguises herself as a man to go to battle. In The Silmarillion, Emeldir the mother of Beren was complimented on her willingness to fight alongside the men by being given the moniker "Manhearted", as if courage in physical confrontations belongs naturally to a man, and not to a woman, and women who can defend themselves and their people are thus by definition man-like.
Galadriel, "mightiest of Elven women", was nicknamed "Man-maiden" in her youth for her excellence in athletics. Yet again, Tolkien implies that physical strength in women is a "mannish" trait. Such views are of course not limited to the British, so one could say that Tolkien was only reiterating a common cultural opinion which is held around the world.
But even if a woman achieves the ultimate 'male' honor of slaying a monster in battle, she still does not have the respect which is given to men who have done less. Eowyn, even after accomplishing a feat that no man could do, was still described rather condescendingly in the chapter The Steward an the King. There, she meets Faramir, and within a few moments of conversation with the man, quickly acknowledges her mental inferiority: "She guessed that this tall man, both stern and gentle, might think her merely wayward, like a child that has not the firmness of mind..." Although Faramir is 12 years older than Eowyn, the woman was already 24 at the time she slew the Nazgul, and definitely not a child. There is little justification for the author to project this condescension onto the character by making her self-deprecate in such a manner.
Tolkien seems to mention a woman's looks at the most inappropriate times. When Eowyn faced the Nazgul, what inspired Merry to aid her was her beauty - "She should not die, so fair, so desperate".1 Even when a woman is fighting for her life, her beauty is relevant. It even matters when she is supposedly dead. The Prince of Dol Amroth comes across the Rohirrim bearing Eowyn, who was presumed dead after her battle with the Nazgul: "the prince seeing her beauty, though her face was pale and cold, touched her hand as he bent to look more closely on her. 'Men of Rohan!' he cried, 'Are there no leeches among you? She is hurt, to the death maybe, but I deem that she yet lives.'"2
If Eowyn had not been beautiful, she might not have any aid against the Nazgul, and the Prince of Dol Amroth would have left her for dead.
Tolkien's stories romanticize the idea of a woman who can't live on without her husband. But there are no tales of the man who can't live on without his wife. The story of Rian, the mother of Tuor, is told in Unfinished Tales:
"Rian, wife of Huor... could hear no news of her lord... And Rian said to the Elves: "... I beg of you to foster [my infant son].. but I must go in search of Huor, my lord."
Then the Elves pitied her; but one Annael... said to her, "Alas, lady, it is known now that Huor fell at the side of Hurin his brother; and he lies, I deem in the great hill of the slain that the Orcs have raised upon the field of battle."
Therefore Rian arose and left the dwelling of the Elves, and she passed through the land of Mithrim and came at last to the Huadhen-Ndengin in the waste of Anfauglith, and there she laid her down and died.
So here you have a woman who chooses to die to join a dead husband rather than live and take care of a living child. Tolkien is very particular about genealogy and he always portrays the ancestors of his heroes in a positive light. Rian is an ancestress of Aragorn, the kingly hero of LOTR, and also of Arwen, the noble, wise and beautiful Elf. So I don't think he intends to put Rian's actions in a negative light, but rather show the reader the romantic image of a noble, devoted wife. The flip side is that she is an irresponsible mother and a needy, dependent person who doesn't have the mental or emotion strength to live on without a man. But in the male-centered world view, none of this matters.
The same formula recurs in Appendix A of the LOTR trilogy, which tells of Arwen pining to death after the passing of her husband. An immortal with the accumulated wisdom of many centuries has much to contribute to society by living on, but none of that matters when her man is dead. As with her ancestress Rian, a woman's yet-living children are not enough reason for her to remain alive. Arwen leaves her son and daughters and wanders off by herself to die of grief. Apparently, the husband-wife bond is held much higher than the parent-child bond in such romantic tales.
Such are the heroines of Tolkien.
1. LOTR, p823
2. LOTR, p827