The genesis of this poem is in the fiercely Catholic and masculine culture of 1900's Spain, where Lorca grew up. This was not simply unfriendly to homosexuality, but actively hostile to it. As a young gay man, Lorca would have found it impossible not to absorb some of the cultures around him, and yet it was irreconcilable with his sexuality.
In New York, he found a much more open culture, but one that was also heavily mechanised, in contrast to rural Spain. This was, in turn, problematic for Lorca, who romanticised the countryside. This poem is an attempt by him, then to weave together these two problematic strands of this thinking with his love and admiration for the American bisexual poet Walt Whitman.
To actually answer the question, let's return to the opening stanzas.
By the East River and the Bronx
boys were singing, exposing their waists
with the wheel, with oil, leather, and the hammer.
Ninety thousand miners taking silver from the rocks
and children drawing stairs and perspectives.
This is the mechanised world, which Lorca dislikes. In using sexual imagery here - "exposing", "oil", "leather", the phallic "hammer" - he is conflating it with heterosexuality. This continues over the following stanzas. This is then contrasted with their lack of wonder at the beauty of the natural world or the act of imagination.
But none of them paused,
none of them wanted to be a cloud,
none of them looked for ferns
or the yellow wheel of a tambourine.
And the end result?
New York, mire and death.
The poet then spends several stanzas idolising Whitman, both for his imagination and his physical beauty, imagining that he alone has the power to save New York from "mire and death". Crucially, Whitman is praised in terms that evoke imagination and nature, in contrast to the industrialised city around him. This, again, is deliberately sexual.
Not for a moment, virile beauty,
who among mountains of coal, billboards, and railroads,
dreamed of becoming a river and sleeping like a river
Shortly after, the long section of the poem highlighted in the question begins, which terminates in the call for "no quarter". But although both Whitman and the "faggots" are gay, it starts by contrasting the two, again in terms of nature against industry.
He's one, too! That's right! Stained fingers
point to the shore of your dream
when a friend eats your apple
with a slight taste of gasoline
So: when the poet ends up fulminating against the "faggots", what he's actually angry with is not gay men, but gay men who choose to live in the confines of heterosexual, industrialised society instead of the more sensitive, imaginative natural society Lorca himself wants. This is why they are "slaves of women". By choosing to exist within a masculine, patriarchal, society that does not want them, they occupy the lowest social order, below that of women.
What Lorca is attempting to do here is separate the hostile anti-gay society in which he was bought up from the society he wants without explicitly condemning his boyhood values. He sees and wants gay society and traditional, masculine society to be separate because they venerate and cherish different values. To him, then, gay men who don't share this vision but instead choose the confines of patriarchy, are almost pestilential. The imagery he associates with these "urban faggots" is animalistic and lustful.
This, then, is the purpose of that final stanza:
I want the powerful air from the deepest night
to blow away flowers and inscriptions from the arch where you sleep,
and a black child to inform the gold-craving whites
that the kingdom of grain has arrived.
The "black child" and the "whites" is, of course, another juxtaposition of bigotry, racism this time, with the implicit assumption that, again, the natural "flower" society, the "kingdom of grain" is the one the oppressed should crave. Lorca "wants" a clean, natural breeze to blow away industrialised society, making space for the oppressed to enjoy a more open, imaginative and playful culture away from the "gold-craving" that drives so much of Western culture.
- Beauty and The Beast: Homosexuality in Federico García Lorca's "Oda a Walt Whitman". Ruth, Tobias. UCLA Department of Spanish and Portuguese, Mester, Volume 21, Issue 1. 19