The partisan divisions over judges are not Harry's fault. Blame Ike. He picked Earl Warren, who led the Court into what conservatives saw as upsetting hallowed tradition (Brown, Baker v. Carr), favoring Communists, coddling criminals (Gideon and Miranda), removing religion from public life (Engel), etc. etc. I believe all of Volokh.com are too young to remember those days, but "Impeach Earl Warren" was a war cry of the right, dating back to the late 1950's. Warren was even under attack by the ABA leadership (then a stalwart of the right).
That was part of the context for LBJ's selecting his successor (Warren got cute, too cute, by resigning in 1968 dependent on a successor, thus giving LBJ the chance to name the successor--LBJ named Abe Fortas):
This source provides a summary of Warren's career, including this discussion of the result:
" On July 11 the Senate Judiciary Committee opened hearings. Shortsightedly, Fortas accepted the Judiciary Committee's invitation to testify. He was the first CJ nominee in history ever to do so. Warren had declined to do so, letting his record speak for him. Senators could not impeach Warren, but they whiplashed Fortas for cronyism with LBJ and drew him into a discussion and defense of several Warren Court decisions, many of which were decided before Fortas had joined the Court. On September 13, the embattled and embarrassed Fortas wrote Mississippi Senator Eastland, the anti-Warren Court Judiciary Committee chairman, that he would no longer testify. The full Judiciary Committee finally approved the Fortas nomination 11-6, moving it to the Senate floor. A coalition of southern Democrats and Republicans, urged on by Nixon, began a filibuster which could not be stopped by sufficient votes for cloture. Fortas wrote to LBJ requesting that his nomination be withdrawn. Immediately LBJ complied with deep regret. Everybody then besieged LBJ with a new nominee, but the proud old Texan refused to name a Fortas substitute. "
Rick Shenkman at POTUS writes:
" Any number of turning points could be selected. One that stands out was the 1968 fight to stop LBJ from naming Abe Fortas chief justice of the Supreme Court. Nothing like it had ever been seen in American history by one measure. For the first time ever opponents staged a filibuster to block a president's nomination to the Supreme Court."
The Post covers the Fortas filibuster here:
Ralph Luker at Cliopatria discusses Blackmun and incidentally agrees that Warren and the Fortas fight were keys.
The Roe decision was significant because it was the Burger court, thus undermining any belief Republican appointees would reverse the direction of the Warren court was undermined, and it was a hot button issue for a new group. Where the hardhats had seen the court as pinko, now their wives became involved.