A vocal minority of "pro-life" Dems were openly hostile to enshrining abortion rights into law under the Patient Protection and Affordable Care Act of 2009. Then a handful of "pro-life" Dem Senators killed an attempt to codify the tenants of Roe v Wade into law in 2017. Finally, the current President has refused any effort to add more justices to the SCOTUS in order to marginalize the sitting Conservative Catholic Califate dominating the branch.

So, on at least three notable occasions, yes. Dems have run away from abortion rights as an issue, while winking at their constituents and whispering "We're the only ones who will protect you."