Right now you need 2/3 vote in both Houses. That won't happen under current conditions.
As for claiming they changed their minds, one could present their history of saying they wanted to overturn Roe. I don't think the Senate investigated seriously the last 3 justices for their anti-Roe views. The Democrats followed tradition of voting to install the SCOTUS regardless of who appointed them. Yes they balked at Bork, and maybe there were others I'm forgetting. It's moot now unless impeachment becomes a serious option.
Bork is the one that people remember, but others have failed to get the nod over the centuries for various reasons. Bork is the most recent one to get outright rejected however, as they didn't vote on Garland and Meirs withdrew her nomination.