This is not, as the quote indicates, begging the question. There is no assumption, as that quote implies, that the fetus is part of the woman's body, only that her blood is. Asserting that you don't get to use my blood supply without asking me first is not claiming you are a body part of mine.
The famous argument by Judith Jarvis Thompson comes in here -- if someone randomly attached themselves to you as a Siamese Twin and made themselves dependent upon your blood supply, would you have the right to detach them? (Especially if you knew that the ultimate outcome of their remaining attached to you would be quite painful.) How much force would you be permitted to apply to get them detached? At what point would inaction on your part tacitly indicate consent for this to continue?
The argument is still invalid. There are not rights, in the ethical sense, that only some people have. And men demonstrably do not have a similar right to control their own bodies.
Most obviously, men can be forced to use their bodies as weapons against their will and at significant risk of death. The draft existed. It displaced men against their will, and co-opted all of their medical care, putting things into their bodies. It exposed them to dangerous chemicals and psychologically damaging environments and occasionally killed them. That would be not having control over their bodies. Freedom from this is clearly not a right -- we have maintained the mechanism of registration.
But in fact none of us have this right, which is clear for other reasons. If we all really have this right, we have to question the legitimacy of not just the draft, but all public health measures like mass vaccination, all incarceration, even all care given to someone who arrives at a medical facility unconscious. This is just not a right, unless we choose to reconsider all those things we would need to change in order to make it one.
There is a right that covers this -- freedom from unwarranted search and seizure. The main difference is that this right is subject to the consideration of the consequences to others. It is not nearly as absolute as this newly-created right to bodily integrity claims to be. Suspending it can be warranted.
You cannot intrude unreasonably into someone else's life unless you have a good reason -- like solving a crime or preventing an invasion. It is debatable whether that includes keeping a potential child alive, but the courts (in a kind of weasely compromise) have decided it does not, at least not until a given level of 'potential-ness' is passed.
Roe vs Wade ignores the fact that this is about the woman's body, since that is not a specific right, and frames it as being about controlling her actions -- forced service is a form of 'seizure'. They have decided that by American standards, it is unreasonable to find out something about someone and then to force them to handle the important related events in their lives in a given way, unless you can be quite certain you are saving a life.
It is not clear that a child is highly likely to actually result from a given pregnancy. This especially not clear during the first 12 weeks, when the decision forbids interference. It is pretty clear that no one can force you into their service to prevent a 70% chance that a life will be lost, and that is about the odds a pregnancy at that point will run to completion. Our liberty is more complete than that.
At much higher odds, intervening in someone else's life to protect a third party becomes a reasonable choice for the state, and abortion after the point you get to higher odds of completed pregnancy is left to local jurisdiction. After the odds have reached something like 1 in 200, a doctor needs a compelling reason to perform an abortion everywhere in the US.
We cannot make up new rights and decide they apply to only one sex.
