Sen. Susan Collins (R-ME) has said that Judge Kavanaugh told her he agrees with John Roberts on Roe v. Wade: It's "settled law".
Is this another episode of "how gullible is Susan Collins"? Almost certainly yes. But it also offers an opportunity to at least a plausible avenue whereby Roe could not formally be overturned but could functionally be killed off.
Justice Kennedy provides a model. He was part of the trio of Republican-appointed justices who "saved" Roe v. Wade in Planned Parenthood v. Casey. Also in that case, he and a court majority upheld virtually all of Pennsylvania's substantive restrictions on abortion rights (striking down only the spousal notification requirement). And after Casey, Justice Kennedy continued to vote to permit pretty much any abortion restriction that presented itself to the Supreme Court even as he never came out and said "Roe v. Wade is overturned."
It's easy to imagine a similar trend basically eroding Roe into dust. Casey and Roe each offer rhetoric for the Court to latch onto. Roe explicitly acknowledged that the value of protecting fetal life was an important governmental interest. Casey, for its part, allows abortion to be banned at any point prior to "viability", and advances in medical technology have steadily pushed that date back. Just as Casey announced a new gloss on Roe while still reaffirming its "core holding", it's not hard to imagine a Kavanaugh-centered court suggesting that Roe is intact so long as some women in some states (generally, liberal states where abortion rights are democratically-entrenched) can access it, and that there is no conflict with Roe or Casey's "core holdings" when women can still access the morning-after-pill.
Basically, what'd we'd get in this world is a studied avoidance of actually overturning Roe while still permitting various state-level restrictions which make abortion functionally impossible to obtain. The net result is a world that's observationally equivalent to Roe being overturned, but Roberts and Kavanaugh get to pat themselves on the back as respectful of precedent.
And the thing is -- this sort of move is a John Roberts special. Yes, the Roberts court isn't afraid to explicitly overturn precedents when it has to (e.g., Janus or Citizens United). But what it really loves to do is "distinguish" precedents it dislikes in ways that virtually obliterate the old holding. Gonzalez v. Carhart, upholding a federal "partial-birth abortion" ban while somehow not overturning Stenberg v. Carhart (which struck down a virtually identical state ban), is maybe the keynote example of this move.
I still think the most likely result of a Kavanaugh confirmation is Roe gets (formally) overturned. But the Roberts Court has ample tools at its disposal if it wants to bury Roe without actually killing it. The idea that conservative jurists actually need to utter the words "Roe v. Wade overruled" to effectuate their abortion agenda is almost certainly a myth.
UPDATE: Okay, so it turns out Leah Litman did this, but better.