They can call it what they want, but the women's lawsuit calling for abortion on demand has nothing to do with Judaism.
They were not calling for abortion on demand, but rather IVF on demand.
When Roe was overturned, there was concern about how this might affect IVF, a technique used by couples who can't conceive naturally. In IVF, a woman's egg and a man's sperm are combined in the lab and the fertilized egg is allowed to develop for 3-5 days, after which it is transferred into the woman's uterus and develops naturally into a fetus. As this procedure has been refined over the past 30 years, there are at least two stages that raise concerns for IVF doctors and patients.
Rather than removing just one egg from the woman, her body is stimulated to produce 10-20 eggs at one time, which are removed and fertilized in the lab to produce 10-20 embryos. Usually one embryo is examined and transferred to the woman, but if the examination shows defects, or if the transfer doesn't "take", then another embryo is used for the next attempt. Usually multiple attempts are needed, but there are likely to be some embryos remaining. Even after the baby is born, the couple might choose to store leftover embryos for the chance of another child in a year or two (Storage costs about $1000/ year.)
Then what happens to the remaining embryos? AFAIK halachic decisions have permitted their passive destruction (that is, they are removed from cold storage and allowed to dehydrate and "die" naturally), since they are in that very early <40 day period, and have no human-like appearance, that is, they look like a ball of cells.
Most other states that created new anti-abortion laws did include a clause to say that the anti-abortion law is not meant to apply to IVF, but the state of Kentucky did not. Their claim that "life begins at fertilization" seems to disallow destruction of those embryos. Some argue that the technique shouldn't be used at all, or if it is used, the unwanted embryos should be transferred to another woman who wants to become pregnant.
This is opposed by most rabbis. Not using the technique means that these couples will not reproduce, and reproduction is desirable under Jewish law. And transferring their embryos to someone else runs the risk that their child will one day meet someone ("We're so much alike!") and marry them without realizing that they're biological siblings. So in this case, I think it's accurate to say that Jewish law doesn't accept the implications of "life begins at fertilization".
A second stage where there could be conflict occurs when a woman hasn't become pregnant after the first attempts of embryo transfer, and the doctor transfers 2 embryos at one time, in the hope that at least one will implant. This often works, but in some cases 2 embryos both implant. While the infertile couple might be delighted to have a large family, experience shows that when this happens, there's a higher chance that the woman will miscarry or deliver prematurely, risking the life of both fetuses. So it is often recommended that only one fetus be kept alive, and "selective reduction" takes place at around 10-12 weeks after fertilization.
This is closer to abortion, and seems to be disallowed by the state of Kentucky's bill, but AIUI, rabbis do allow this because it significantly increases the chance for a healthy baby.
There are many women and doctors who have these concerns about the impact that the law will have on IVF, but these women are able to base their argument on religious freedom.
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