The support from the right is virtually unanimous and the loathing from the Left equally so. The Left will base their opposition on one ruling - Casey.
Alito is best known for casting a dissenting opinion in Planned Parenthood vs. Casey (1991). While the Left will aggressively sling Casey as napalm, Patterico gives a very good narrative of what Alito was getting at here: An excerpt:
Judge Alito then noted that the spousal notification provision at issue did not give the husband a veto power. Rather, a married woman simply had to certify (through her own uncorroborated and unnotarized statement) either that she had notified her husband, or that her case fell within any one of several statutory exceptions, including:
(1) [The husband] is not the father of the child, (2) he cannot be found after diligent effort, (3) the pregnancy is the result of a spousal sexual assault that has been reported to the authorities, or (4) [the woman seeking an abortion] has reason to believe that notification is likely to result in the infliction of bodily injury upon her.
Judge Alito then argued that the appellees challenging the statute had not met their burden of proof — which Justice O’Connor had said rested with those asserting an “undue burden” — to show that the law had the “broader inhibiting effect” required by the whim of Justice O’Connor the relevant precedents.
Alito noted that the evidence showed that 1) most abortions are sought by unmarried women, and 2) about 95% of married women seeking abortions tell their husbands in any event. Of the small number of women remaining, the record was devoid of evidence as to what percentage would be unable to assert at least one of the four exceptions enumerated above. Nor was evidence offered to show what percentage of women would suffer retaliation from their husbands in ways not covered by the four exceptions.
Judge Alito concluded that, absent any evidence as to how many women would be adversely affected, the appellees had failed to meet their burden of showing that the spousal notification requirement imposed an “undue burden” on women. He specifically noted:Whether the legislature’s approach represents sound public policy is not a question for us to decide. Our task here is simply to decide whether Section 3209 meets constitutional standards.
The key here is that Alito, referencing Sandra Day O'Connor's previous SC opinions, felt that requiring a woman to sign a non-notarized and uncorroborated document indicating that she had notified her spouse before having an abortion did not pose an undue burden. There are four, fairly broad exceptions - including risk of bodily harm - to this requirement.