It will be really interesting to see what a Republican majority lets happen here, and I’m not sure there’s a good way to tell. There has never been a situation in the post-Nixon era where a GOP Senate majority confirmed a Democratic appointee to replace a Republican appointee on the Court.
Frist will be in charge when Rehnquist dies but McConnell is already waiting in the wings. I think some of them may shrug and say “well that’s what happens” but McConnell won’t — he’ll demand that Kerry appoint someone previously appointed to a lower court by a Republican. Will the caucus side with him?
You could see a situation where Frist allows Kerry to appoint a moderate like Garland in exchange for a slew of lower court nominees who are Fed Society approved but idk if even that would be good enough for McConnell
Pre-2016, much of the of GOP bad blood towards the Democrats on court fights was a product not only of Bork, Thomas, and Alito, but also a sense that their agreements with Democrats were never adhered to. Democrats stonewalled lower court nominees until 2005 (right after W's re-elect), the GOP cut a deal with the gang of 14 to avoid nuking the filibuster, the Democrats went back to stonewalling nominees after they took the Senate in 2006, the GOP retaliated in kind from about 2009 to 2013 (right after Obama's re-elect), and the Democrats responded by nuking the filibuster themselves after catastrophizing about how bad it would have been if Republicans did that. Reid left the Supreme Court filibuster in place, but a great many Democrats like Schumer and Obama had already tried to filibuster Alito.
My overall point with the above paragraph is to note that the GOP of 2005 is going to be a lot less willing to play hardball with the Democrats than the GOP of later years was. Compromise probably wins out over obstruction.
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Second, George W Bush - faced with the piling up headaches of Iraq being a mess, the botching of Katrina, and other problems I can't totally recall, basically re-nominated the already well-liked Roberts to the Chief Justice spot (he was first nominated to O'Connor's seat) because of a desire to avoid a distracting Court fight. Kerry will have the complications of
(a) being a new President transitioning in and staffing a whole new Cabinet;
(b) most of the Iraq problems W dealt with; and
(c) Katrina, which even if handled more competently than W will still inevitably be a massive problem since local and many logistical factors will remain the same no matter who is in charge. I don't see why Kerry - who doesn't quite have a massive mandate given the GOP is holding the Senate and he might be winning the Presidential election without winning the popular vote because 2004 really was that close - would be any more eager for Court fights than Dubya was.
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In all likelihood, an ATL version of the Gang of 14 occurs where Kerry reappoints many of the the Judges the Democrats signed off on OTL, but withdraws the ones who weren't signed off on.
If the fight over the Chief Justice appointment and O'Connor replacement drag out, it's possible that O'Connor just rescinds her retirement. The point of her retiring OTL was to be with her husband before he was mentally gone due to Alzheimers, but by the time she left the bench (after delaying her retirement a year so as to not leave the Court with two vacancies at once) it was too late. OTL she seemed to regret her retirement - not only because she disliked Alito (though she pretty much adored Roberts but detested Alito) but because she retired from a job she loved to spend time with her husband who was a key part of getting her that job, and he was gone by the time she stepped down.
If Kerry made O'Connor Chief, he'd get some liberal brownie points for naming the first woman Chief Justice even if it wasn't a massive victory for liberals.
Garland seems like an easy appointment to the Supreme Court - and sort like a liberal version of John Roberts. Mostly interested in legal issues rather than bigger cultural stuff, for example. Compare AG Garland's response to Dobbs (which was a list of ways the current Supreme Court likely would sign off on how to protect abortion) with the Dobbs dissenters (a long angry tirade, without trying to give guidance to lower courts on how to minimize the damage the Court just imposed on the country) and it's a pretty stark difference.
Brian Sandoval might be a possibility for the Supreme Court. Harry Reid basically got Bush to put him on the District Court in 2005 to prevent him from running for office down the line (Sandoval ultimately stepped down from the District Court and beat Reid's son for governor in 2010). Reid tried to convince Obama to nominate him in 2016 too. The criticism would be that he doesn't have judicial experience, but Reid (among others) seemed to view this as a positive thing - he thought the Supreme Court didn't have enough diversity of experience on it. He supported Harriet Myers, for example, on the grounds that she had experience in private practice. Sandoval would be friendly to liberals on issues like abortion and environmentalism, most likely.
Jose Cabranes would be a moderate choice and a pretty hard guy to oppose. His background was over the top in almost every regard.
Leah Ward Sears is a possibility. She was a Zell Miller appointee to the Georgia Supreme Court, fairly liberal, and personal friends with Clarence Thomas. She was on Obama's list in 2009/2010 too.
If abortion, LGBT rights, and gun control are the most important issues - Richard Posner is a guy the Democrats might want to nominate. The problem is he basically doesn't believe civil liberties or human rights exist as a more abstract matter, is pretty firmly right-wing economically, and has a record of public statements and papers such that he could pretty easily be painted as a sociopathic lunatic.
Sidney Thomas of the Ninth Circuit is another name, as is David Tatel who has been mentioned previously in this thread.
Diane Wood would be a brilliant liberal appointee, having tested her mettle against Posner and Easterbrook.
I don't have much to add beyond
@oliveia's post on the New York Times list. Of the names on that list, Kagan would be the strongest stealth-liberal and compromise-broker. Lawrence Tribe OTL basically said to Obama nominate her or Diane Wood, because Sotomayor - while Liberal - won't be good at wooing the right-wingers towards the middle.
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My guess is Kerry gets to replace Rehnquist, O'Connor, and Souter. Stevens would be too stubborn to quit until he starts slurring his words (which, according to Stevens, wasn't until the Citizens United case in 2010), but maybe he retires. If Stevens retires, it would be after the 2006 elections (when the GOP will in all likelihood have over 60 seats in the Senate I think, but others in the thread seem to disagree). But it's also possible that in all of the tumult, O'Connor decides to stay on the Supreme Court.
The GOP will make Kerry's transition and staffing a new administration a bit of a headache, and I can imagine various Democratic Senators getting appointed to Cabinet posts ... which would mean they aren't in the Senate to vote for Supreme Court nominees. Likewise, even if Romney is obliged to nominate a person of Kerry's party or from a list presented by the legislature ... Romney could drag his feet and use the opportunity to build up his own image as a guy obstructing the Democrats. Romney was already intent on running in 2008 OTL, so a high-profile fight seems like something he'd relish. And if the legislature requires that the seat be left vacant until a special election takes place - that could be four months or more of an empty Senate seat, which is bad for John Kerry. And I'm not entirely sure the Democrats would win that special election, given that Scott Brown won in 2010 OTL.
Overall, my sense of the Kerry Administration is that it'll end up as a kind of placeholder Presidency overwhelmed by a mix of cleaning up after Dubya's mess, by not having majorities in Congress, and by being affected by lots of stuff beyond its control which leaves it too busy to have a clear goal in mind of what it wants to do. From the Administration's point of view, it'd probably be Four years of one damned thing after another.