- The episode is taking a strong anti-abortion, misogynistic stance.
- The episode is pro-choice, and emphasizing that pro-choice means pro-choice, in favor of people having options and making their own decisions.
- The episode isn't about abortion at all and people are getting upset over nothing. As in most debates, this seems to be the angriest group.
However, there are counterarguments to both the arguments I just mentioned. First, late-term abortions are performed when the pregnancy is life-threatening, and there is a risk that the destruction of the Moon will cause serious damage to the Earth and its people, so neither of the arguments I cited in the previous paragraph necessarily hold. This leads to the second read, in which the creature is a fetus being incubated by the Earth, putting the people of Earth collectively in the position of its mother. They choose to abort, and are overruled by Clara, Courtney, and the scientist (was she even named? I never caught it if she was), who act in the position of the anti-abortion government and force the Earth to carry through the risky and difficult labor. The strongest support for this position are Courtney's repeated declaration "It's a baby!" and refusal to even consider killing it as an option, and Clara's teary rejection of being given a choice as patronizing. Read this way, the episode is pretty clearly repeating the misogynistic arguments of anti-choicers, denying the agency and autonomy of the collective mother in favor of the moral judgment of a tiny minority.
But there are arguments against this reading, too, the biggest being that the first read makes both Clara's final confrontation with the Doctor and her ensuing conversation with Danny afterthoughts, while the second badly misunderstands what Clara is saying in that confrontation--she's not angry that the Doctor left her to make a choice on her own, she's angry about him denying her information that would have been useful in making her choice and his general condescending attitude. Which brings us to the third way to understand the relationship between the three players: to see the hatching creature as a fetus threatening the well-being of the women on the moon, and the people of Earth as voters in a democratic government that nonetheless lacks the moral authority to tell those three women what to do in regards to a choice that impacts them directly. In this read, the episode is emphatically pro-choice, as it says not even a popular vote with (note the visual pun) one hundred percent turnout can legitimately tell a person what to do when their bodily autonomy is on the line.
And I do think this is the strongest reading, for a number of reasons. One is that visual pun, and another that shows up a bit later: the button Clara and Courtney press to cancel the bombing is, as indicated by the bombs' display, the ABORT command. Further, the women standing around the bomb fall rather neatly into a couple of archetypes involving trios--and yes, I know I've been rather hard on archetypal readings in the past, but that's because I find they are often used in constraining and limiting ways; I'm all for making characters serve as multiple contradictory archetypes simultaneously or otherwise using them in ways that multiply, rather than restrict, available readings.
One way to read the trio of women is as representing Women with the tired old "triple goddess" routine. They slide quite neatly into the roles: Courtney is a child, and hence the Maiden; Clara is the next-oldest and takes care of children for a living and the possibility that she has children on Earth in 2049 is floated, making her the Mother; the scientist is the oldest of them, starting to show wrinkles, and emphatically rejects the notion of having children, making her the Crone. It's a reductive and excessively reproductive way of defining Women, but perhaps a reproductive approach is not entirely inappropriate when discussing reproductive rights.
Contradicting and coexisting with that read is one in which they represent a single woman via Freudian nonsense: "disruptive influence" Courtney serving as the id, the scientist--professionally rational and the one endorsing going with the judgment of society as a whole--as the superego, and Clara mediating between them as the ego. Again, a silly way to construct a person, but still a read reasonably well supported by the episode. In this read it's interesting that the "It's a baby!" attitude is associated with the unreasoning, over-emotional aspect, while "I fail to see the moral dilemma" is associated with the most rational aspect. This is pretty accurate where the abortion debate is concerned. Admittedly, the waters are somewhat muddied by Clara's decision not to destroy the creature, but again, that's because in this read the episode is attempting to navigate the nuance between being pro-choice and rejecting the anti-choice narrative that pro-choice activists are genocidal fetus-haters.
Ultimately, though, all three of these readings seem reasonable and supported by the text, as I'm sure are a multitude of others. This is a justified controversy, and I think a good one, as it seems likely to, in among the usual fractious debates, produce some conversations worth having.