There are lots of things that are generally accepted as harmful to a developing fetus, such as drinking alcohol, lots of caffeine, smoking, etc.
Would a pregnant woman who did something known to cause harm to the fetus be open to being charged with assault or something similar?
For example, suppose a woman gets pregnant, doesn't stop drinking, and the baby is born with fetal alcohol syndrome or some other pathology. Is that considered a crime?
Yes or no, depending. The question is investigated in "The Status of Pregnant Women and Fetuses in US Criminal Law" (JAMA), which collects 23 opinion in US jurisdiction. In Reinesto v. Superior Court, 182 Ariz. 190 where the court ruled that the state cannot prosecute for child abuse a woman who uses heroin during pregnancy and thereafter gives birth to a heroin-addicted child. However, in Whitner v South Carolina, 492 SE 2d 777 the court did find the mother criminally liable for child abuse, based on prenatal drug use. The South Carolina case is the sole example of that type, in the study (published 2003).
Charges range from child endangerment/abuse, illegal drug delivery to a minor, or fetal murder/manslaughter. The general finding is that since a fetus is not legally deemed to be a person (in those jurisdictions, at that time), where was no "child abuse". South Carolina, on the other hand, reasoned, here and in prior cases, that
We have no difficulty in concluding that a fetus having reached that period of prenatal maturity where it is capable of independent life apart from its mother is a person.
A different study ("Criminal Charges for Child Harm from Substance Use in Pregnancy", JAAPL) which included cases up to 2015, found a slightly different distribution but generally concludes that courts do not consider maternal drug use to be a legal question. That article also cites a web page which at the time is purported to say that 18 states allow civil child abuse proceedings.
In Chenault v. Huie (Texas), the court found that
Texas does not recognize a cause of action in tort for injuries to a child that result from the mother's negligent or grossly negligent conduct while she was pregnant with the child
but according to the more current Guttmacher Institute study, about half of the states have such a civil cause of action.
Is that considered a crime?
NO, not if the baby lives
The leading case is CP (A Child) v First-Tier Tribunal (Criminal Injuries Compensation) [2014] EWCA Civ 1554 in which compensation was sought for a child on the basis that the mother's drinking was an offence under section 23 Offences Against the Person Act 1861:
Whosoever shall unlawfully and maliciously administer to or cause to be administered to or taken by any other person any poison or other destructive or noxious thing, so as thereby to endanger the life of such person, or so as thereby to inflict upon such person any grievous bodily harm, shall be guilty of felony, and being convicted thereof shall be liable to be kept in penal servitude for any term not exceeding ten years.
However, the Court of Appeal found that a feotus is not a "person" within the meaning of section 23 so this offence is not committed in these circumstances.
The publisher Family Law offers this detailed summary:
The child was born with foetal alcohol spectrum disorder as a consequence of the mother’s excessive drinking during pregnancy. An application for compensation was made to the Criminal Injuries Compensation Authority. The application was rejected on the grounds that the child had not sustained an injury directly attributable to a crime of violence within the terms of the Criminal Injuries Compensation Scheme 2008 [CICA1].
...
YES, MAYBE if the baby dies as a result.
I cannot find and caselaw for a mother being liable for her baby's death in this way, but in a related case, the Attorney General's Reference (No 3 of 1994)([1997] 3 All ER 936, as well making it clear that a foetus was not to be regarded as another person, injuries to one in the womb that causes death following a live birth would be manslaughter - the offender in this case stabbed the mother which injured her feotus who died soon after birth.
1The Criminal Injuries Compensation Authority (CICA), is a government funded organisation that was created to provide compensation for blameless victims of violent crime. A conviction is not necessary, just enough evidence to meet the criteria for compensation.
Though there's no federal law banning the use of alcohol while pregnant, a few states consider drinking alcohol while pregnant child abuse:
Some of the relevant statutes are cited here.
Substance (drug) abuse is even more restricted, with about 24 states and DC having specific laws regarding doing it while pregnant.
As for tobacco, I couldn't find any statute banning the practice.
