Presidents propose, Congress disposes. This ain't going nowhere, and wouldn't even if Trump wasn't such a wounded duck.
Eh, this is something that Democrats could get their caucus to unanimously support if they wanted to. So all you need are 30ish moderate R's in the House to vote to add into the next deal to keep the govt funded. As far as the senate goes, Heller and Collins would be crazy to oppose this and Dems would be crazy to filibuster (if the filibuster applied to the rules for whatever it was attached to). That means they just need one of Gardner, Tillis, or even Flake or Perdue to get nervous about backlash from suburban women in 2018/2020. Even though she is electorally safer than them, Murkowski is probably the 3rd best target for this.
I think this is modest enough to pass if attached to a budget deal and if Ryan and McConnell let it come to the floor. Something like 1 yr primary caregiver, 3 months other parent would be DOA though. Which raises an interesting point: Why didn't Dems do this in 2009? They could probably have gotten 12 weeks back then if they wanted to try.