As soon as the appointee is appointed by a Democratic politician, at least.

That won't help. The count is currently 6 Republicans to 3 Democrats. The party isn't an absolute predictor of how they'll vote, but it's pretty close.

It would take a minimum of two Republicans to retire, and be replaced by Democrats, to change the partisan balance. The oldest justice is Clarence Thomas, but the next two are both Democrats. Everybody else is under 70 and will not retire for a minimum of a decade.