The argument comes down to when life begins, and we all know that isn't declared in the Constitution, nor (as far as I know) in the US Code of Laws.
But is it the job of the government to establish an official Starting Time for human life? It is a moral, ethical, and religious question, and while some religions and schools of thought believe that life starts at the instant *snap* of conception, others believe that it does not begin until the actual birth occurs, and there are a zoo others that place it at just about every step in between. To have our government decide which belief system or religion is correct is problematic on several levels, and would open a Pandora's box of unintended consequences.
Our government is instead charged with the responsibility of protecting everyone's ability to choose for themselves. Since the courts must therefore set aside the charge of trying to decide when it is or isn't too late for an abortion, their decision must be a medical issue on the part of the mother. That's the reasoning, step by step, why the courts should guarantee the right to every woman's privacy and bodily autonomy.