Some of the state laws banning abortion in the US contain exceptions for health reasons, typically where the pregnancy threatens the health of the mother or if the foetus is not viable.
There have been cases where women seemed to qualify for an abortion under one of these exceptions, but were denied an abortion by medical staff on the grounds that the medics could not be certain that the police or courts would agree.
For instance, one woman who's foetus had an incomplete skull was denied an abortion:
Because Louisiana’s list of conditions justifying an exception from the state’s abortion ban did not explicitly include acrania, hospital officials turned down terminating Davis’ pregnancy, apparently fearing they could be subject to prison time, costly fines and forfeiture of their operating licenses if they performed the procedure.
The relevant exception in this case is for a "medically futile" pregnancy, defined as:
(4) "Medically futile" means that, in reasonable medical judgment, the unborn child has a profound and irremediable congenital or chromosomal anomaly that is incompatible with sustaining life after birth. This diagnosis shall be a medical judgment certified in the pregnant woman's medical record by a reasonably prudent physician who is knowledgeable about the case and the treatment possibilities with respect to the medical conditions involved.
Likewise the same law also allows for abortion when
necessary, to the best of that person's reasonable medical judgment, to prevent the death of the pregnant woman or to prevent a serious risk of the substantial and irreversible impairment of a major bodily function of the pregnant woman.
This requires a doctor contemplating offering an abortion to decide whether the risks to the mother are "substantial" and "serious". Again, people of common intelligence might well differ about any individual case. Could it be that exceptions like this render the laws unconstitutionally vague? The vagueness doctrine is:
[T]he terms of a penal statute [...] must be sufficiently explicit to inform those who are subject to it what conduct on their part will render them liable to its penalties… and a statute which either forbids or requires the doing of an act in terms so vague that men of common intelligence must necessarily guess at its meaning and differ as to its application violates the first essential of due process of law.
It would seem, as a practical matter, that the medics in the Louisiana case above were genuinely uncertain as to whether administering an abortion would render them liable to severe legal penalties. They were uncertain whether police, prosecutors and jurors "of common intelligence" (i.e. not trained in medicine) might consider them "reasonably prudent" after the fact if they carried out an abortion.
Could some or all of an anti-abortion law be struck down for vagueness as a result?
I know that this is asking about a hypothetical case, and there hasn't been time for any US court to make judgements about such a case. But I'm interested in whether such an argument would stand a reasonable chance in a court.