Alito’s draft opinion already alludes to - in his phrasing, an unenumerated right (like abortion) can only be protected by the courts where they have a strong basis in the history and traditions of the country.
He specifically mentions both Lawrence v Texas (sodomy) and Obergefell v. Hodges as being likewise poorly decided. The only thing that would perhaps save those rulings would be one of the five who agreed to overturn Roe with him balking on those, which I would not count on.
So ... Obergefell v. Hodges essentially suffers from the same “defects” Alito found in Roe, and would seem a hot candidate for being overturned. Griswold v. Connecticut rests on similar concepts.
If Griswold goes, then privacy largely goes with it, it will remain to be seen if Roe can go without Griswold going. Many smart legal minds have long said it would be very difficult to overturn Roe but uphold the privacy conclusions that essentially made Roe an obvious decision, and those come from Griswold, without eventually overturning Griswold as well. Otherwise you are staking out that Griswold establishes a right to privacy for everything other than abortion, which suffers many of the logical defects Alito asserts about Roe. The reason you will struggle with this regime legally is because it is hard to enforce many laws against abortion if there is
a robust right to privacy, because the State has to intrude fairly far into your privacy to enforce those laws. Once the right is undermined, how does Griswold stand? Without Griswold how does privacy as a constitutional right stand?
There is also a movement for a national abortion ban. Major antiabortion groups have already spoken with Trump and other top Republican Presidential contenders in 2024 and all of them say they would have no problem running with a nationwide abortion ban bill as a centerpiece of their campaigns. 6 in 10 Americans are against Roe v. Wade being overturned, but the reality is only a small portion of that 6 in 10 votes on the topic, plenty of them support abortion rights in principle, but not enough to make them vote for Democrats.
The filibuster likely won’t protect anything – Republicans will certainly dispense with the filibuster if they control both houses of Congress and the White House, because they will wager they will never again be in the minority so there is no reason to keep a relic of minority power in the Senate.