What is the role of women in the poem "The Lover Song of J. Alfred Prufrock"?

In a word: Perplexing.
Women appear in several places in the poem, both directly and indirectly. He notes that

In the room the women come and go
Talking of Michelangelo.

In context, the women in these conversations seem vapid and lofty. Yes, they can engage in superficially educated conversations, but they "come and go," not staying to discuss the art deeply. Michelangelo is a well-known artist with whom most people are marginally familiar, at least. Therefore, this reference seems especially unflattering of women, especially when considered in the context of the previous stanza. The speaker and an unnamed "you" have had restless nights spent in cheap hotels and have argued with "insidious intent."

Later the speaker blames the vices of women for his frequent state of frustrations:


Is it perfume from a dress
That makes me so digress?



The smell of a woman causes him to feel settled. He notes that his days are passing one by one, one coffee spoon and one afternoon at at time, and women offer nothing in the variety of life:


And I have known the arms already, known them all—
Arms that are braceleted and white and bare



One woman is the same as the next. There is even a note that they try to deceive him with their white, bare arms with his parenthetical statement that

(But in the lamplight, downed with light brown hair!)

If they are deceptive about their arm hair, what else could they be deceiving him about? They are suspicious creatures, and even the perfumes they adorn themselves with lead him astray.

His concluding stanzas present a hypothetical question:


And would it have been worth it, after all,
Would it have been worth while,



If he commits to one woman and commits to spending his afternoons, porcelain, cups, and novels with her exclusively, will it all be worth it? Or will he forever feel misunderstood in his relationships with women:


If one, settling a pillow by her head
Should say: “That is not what I meant at all;
That is not it, at all.”



The speaker's tone toward women is not one of love and adoration; in fact, he seems to struggle with connecting with women period.