More seriously-I think that not allowing minors to seek abortions or reproductive health care, or parental notification, is extremely dangerous because of what is likely happening that gets a minor pregnant (and it is pretty damn well documented that comprehensive sex ed dramatically reduces teen pregnancy and abortion rates) but that's probably less relevant to the strictly AH question.
O'Connor's position basically amounted to "you can require parents to sign off on abortion for minors as a general manner, but only if in extreme circumstances social workers can go to a judge and have them sign off on an abortion being granted."
Have that win out early on over other circumstances, and that might dampen things.
I don't think any one policy change would prevent the big reaction to Roe OTL. You essentially just need a much narrower Roe which moves the conversation bit-by-bit so society can sort of adjust to it over time, rather than one big swoop. This was the main criticism of Roe of a lot of pro-choice legal scholars who absolutely wanted the Court to strike down the laws that the Court did in Roe v Wade. Lots of people bring up Ginsburg's criticism of Roe, but they miss that Ginsburg's criticism was that Roe did too much too fast, not that Roe struck down the laws in question.
Honestly, there's a pretty strong theory that most of what Roe said was merely advisory stuff for lower courts to play around with going forward, rather than part of the decision itself, and that Rehnquist's 1989 Webster opinion (which would had cut back Roe and its companion case to their facts) was what the Roe justices thought they were signing onto (since that's basically what Burger wrote in the Roe concurrence). Rehnquist in private notes in 1972 basically praised the Roe majority as having written a very nice opinion which he merely disagreed with - very different from the quasi-apocalyptic attitudes he and his allies adopted later on. The father of the right to privacy himself - William O Douglas - had a concurrence in Roe's companion case Doe v Bolton which doesn't really make any sense unless a lot of what Roe said Roe didn't really say.
If you have some of the phrasing of Roe itself be a little bit different, and the Court expands abortion access slowly over time - yes there will be social conservatives freaking out over it, but the 70-80% of the country in between the 10-15% most conservative/liberal will probably just accept the gradual growth of abortion access. It's worth taking into account that protestant southern states generally had more *liberal* abortion laws pre-Roe.
The way to get liberal/progressive progress in a broadly conservative society is to do it inch-by-inch, such that nobody really notices and people have a time to process it and get used to it, even if they disagree. LGBTQ+ rights in the Courts, for example, was a long-slog from 1996 (Romer v Evans) to 2015 (Obergefell v. Hodges), where each single step was taken around the time that a majority of the public supported it. When the Court gets involved in social change, it's usually just cleaning up what remains of an already-socially-overruled public consensus, or pushing the ball ahead by 5 to 10 years. Roe's problem wasn't that the laws in question should have been upheld (they probably shouldn't have), but rather than it did too much too fast. Eventually, there'll be a great amount of progress, and people will wonder why anybody ever opposed it at all.
Tbh, I sort of feel like if the Court took it slowly, Planned Parenthood v Casey is where America would have mostly been on the abortion issue by the time of Planned Parenthood v Casey OTL. For comparison, Loving v Virginia was in 1967, but most Americans (black or white) didn't approve of interracial marriage as a private matter until the 90s (though, I emphasize, they had much more liberal views as a question of whether other people should be *allowed* to have such marriages than they did when it came to the question of whether they *morally approved* of having such marriages themselves).
I guess this is a long way of saying that to avoid abortion becoming something other than a Catholic issue, it seems like you'd need to do a boiling frog type of approach. While lurking online, I've observed various pretty fundamentalist protestants and Christians who essentially say 'I think all abortions are evil, but the evils of not allowing to an extent - such as denying women freedom and liberty and equality - are even worse.' For what it's worth - Bill Brennan was a pre-Vatican II arch-catholic who was very very very pro-life as a matter of personal politics. My sense is that this sort of thing I just described is what the Roe Court was trying to get at, but it did too much too fast.
Also, Roe v Wade gets criticized as stupid way too much. If you read the case entirely, Blackmun basically went the extra mile trying to be meticulous and explain the long complicated history of medicine, the advisory function of the medical profession, the challenges of adopting a legal definition of life, how the states were grappling with this in all sorts of areas (like medical malpractice and inheritance laws), etc. I think the fairest criticism is that the Roe Court tried very hard to explain itself and get a very difficult question right, nobody bothered to read the thing in its entirety, the Court probably overdid it in a single couple of cases, and everybody's been talking past each other ever since.
