Fortas stayed on as an associate justice, and Warren,
‘s seat stayed vacant, who announced his pending resignation in June 1968 stayed on until President Nixon successfully appointed Warren Burger to fill his seat in 1969. The Warren Court had been in its most liberal, activist phase, seemingly ready to declare a right to a minimum income, order inter-county busing for integration, and more. Instead, Nixon appointed four Justices, and the Court has been on a gradual conservative trajectory ever since. The 2016 election may prove to be similarly consequential for the Court.
UPDATE: There are three major differences between the Warren/Fortas situation and the Scalia/TBA situation. The first two favor the Democrats: first, the former situation didn’t leave the Court with only eight Justices for a long period of time because Warren delayed his resignation; and, second, the former involved specific objections to a specific nominee, not a blanket refusal to confirm anyone the president nominated. On the other hand, unlike whomever Obama nominates, Fortas likely could have gotten a majority of the Senate to vote for him, and was done in by the willingness of his opponents to filibuster.