The counter-argument about 'right to bodily integrity' misses a whole bunch of nuances. This post is not researched, it’s just “off the top of my head”.
Problem 1. The concept of human rights is a deontological argument akin to the Ten Commandments. It doesn't state why and on what authority, and for what ultimate end, and on what grounds, human rights even exist. They "just are". While I do not want to say that I disagree with them in content, I disagree with them in form and presentation. For a simple example, compare the right to property vs the right to not starve. Communism, as espoused by the EFF, suggests that the right to not starve trumps the right to property. Indeed, the communist manifesto (as in the one by Marx and Engels) actually states that we must make "despotic inroads" into property rights. So it's not obvious that human rights systems are the best. Moreover, I like to distinguish between provisive rights and defensive rights. The DA stands for defensive rights, e.g. the right of a woman to her bodily integrity, the right of an Afrikaans farmer to keep his 100 ha farm, etc. The EFF stands for the rights of the poor (provisive rights: that if you do not have something it will be provided to you).
Problem 2. It is a false dichotomy. Look. Just because someone has the right to bodily integrity, it does not in fact automatically trump everything else. Sorry for the Reductio ad Hitlerum but - consider the case where you can assassinate Hitler and thereby prevent WWII and 60 million deaths. Would you do it, or would his right to life trump the other 60 million? Is his bodily integrity that sacred? You see, there's no clear easy line over which bodily integrity trumps. Remember the "Beethoven" thought experiment. Poor family, lots of kids, an inconvenient 9th or whatever child is going to be born, and deaf to boot. Abort? Most pro-abortionists say 'yes', and then one pulls the trump card, well, you just aborted Beethoven. You see, it's not that cut and dried.
Problem 3. Socioeconomic. Poorer women will get backstreet jobs, especially in countries that frown upon it legally or for religious reasons, with the result that they're more likely to get horrid infections, death, or failed attempts at the cost of their sexual health (eg mutilation) to no good reason. There's a pro/con analysis to be done here. Obviously in South Africa this isn't an issue, but it is, in catholic and muslim countries (or basically any theocratic-leaning state), which are the majority (populationwise if you add them up). So it's not a simplistic question of "don't you tread on my rights", it's a question of "how, when and where", and will you be worse off afterwards?
I have however heard from a very reliable source (a statistician who works with epidemiology and the relationship between crime, life quality, life expectancy, etc etc)... that in countries, provinces and states which allow abortion, crime is lower. In effect, he said, they're aborting future criminals, since it's largely poorer women who can't afford birth control, and therefore, are unable to effectively raise well adapted citizens. I find this discussion disturbing but I can see that how it is supposed to work. However, I find it hard to not see this as coextensive with giving a death sentence for future potential crimes — which strikes me as rather brutal. Put it another way: it’s pre-emptively killing poor people just in case they commit crimes. Ethically I find that problematic. The excuse, of course, is that it's not a punishment meted out for actions yet to be performed, like precrime in Minority Report, but merely that there's a correlation between legal abortion and lower crime rates (i.e. it's not a causal or moral imperative, just a correlation).
Problem 4. A related problem to point 1. If moral skepticism, which I am sympathetic to, is plausible, that is, that the moral amounts to no more than biological harm, it seems to me that if moral = biological prosperity, and evil = biological harm, then it follows that abortion is evil. Think of all the cases of 'evil'; they all fall under biological harm: rape, murder, armed robbery, assault, etc., all involve someone getting physically hurt. In the case of abortion, like carnivory, some small creature is destroyed for convenience. I think that if it is true that 'evil' amounts to biological harm, plus a prohibition against that, then it follows that abortion is evil. However, since I am in fact a moral skeptic, I don't think there's such a thing as good and evil only biological harm and biological prosperity. As such, I think it is debatably justifiable to abort IFF the potential child will face a life of misery: to wit:
if it is guaranteed by empirical testing to be mentally disabled or physically disabled and therefore have a long difficult life and be denied a prosperous normal life, but rather face one of great suffering (the antinatalist position);
if it is a conjoined twin and unlikely to be separable postpartum, for the same reasons as above;
if it is the result of a rape and therefore will be stigmatised and abused by its parents and family;
if it is the result of incest and will therefore have genetic diseases.
The trouble with these types of justification is that there's a morally problematic way to describe them: Eugenicist. Hitler was also in favour of eugenics. Does this mean that eugenics are wrong? That's a separate debate, but one of the reasons often given by the pro-abortion lobby is that eugenic reasons can trump the foetus' right to life as well. So— if you are in favour of allowing abortion, do you support eugenics, antinatalism, or just a woman's rights? I think you have to commit to one of these positions because you can only justify abortion with one of these positions.
Naturally, I have the privilege of being in a long-term relationship, so my opinions here are moot. I recognise that possibly few women are in a situation where they can rely on support. Hence I do not pass judgment on women who make this decision. I have merely, above, given some of my reasons why I think the "right to life" vs "right to bodily integrity" is an unsophisticated argument.