It doesn't animate conservatives that much anymore, since it's been around forever and they assume that even with more conservatives on the Court, abortion will remain legal, albeit with greater or lesser regulations depending on the state. Liberal voters mostly assume the same thing, after so many false panics during the Reagan era, every time a Republican got to nominate a new justice.
By this point, the culture war is tiresome and irrelevant -- we're moving into a new era defined by populism and anti-globalization, not airy-fairy crap in the social-cultural domain.
Therefore, any fighting against the next Heritage Foundation pick should focus on the nominee's position on populist issues. That was Trump's secret recipe for winning as a Republican at the end of the Reaganite era, when too many voters had grown weary of corporate elitism. If he's not going to deliver much on populism through the executive branch, he should at least do something on that score in judiciary appointments.
If he nominates justices who attack populism, that represents a betrayal of his crucial swing voters from 2016. If he is held to account, then we get a populist-friendly justice, and all's well. If he breaks with populism in his nomination, then it will be out there undeniably for all those Rust Belt voters to see -- both when judging him and the whole GOP in the mid-terms and the 2020 presidential election.
Already in Trump's term, the Court has dealt strong blows to labor unions -- not exactly the best way to keep working-class Obama voters happy for switching their vote to you so that you wouldn't go down in flames like McCain and Romney.
These are the issues that should be dissected when the nomination is made -- not because the Court is going to immediately revisit their decisions thus far into Trump's term, but because the next appointment could last for decades down the line.
They are all the more pressing since the already-weak bubble economy for most people is going to go POP even for the 1% before Trump's term is up. In that deep recessionary context, it will be all the more crucial for the bottom 99% to be able to rely on the new justice to have their back.
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The upcoming SCOTUS confirmation battles will be primarily defined by Trump's status as a disjunctive president, serving in the dominant party of his era at the internally dysfunctional end of that era (GOP during the Reagan era).
The disjunctive phase almost always has the dominant party controlling both the White House and Congress, so that any conflict cannot stem from a partisan mismatch between executive and legislative branches.
Conflict will only be due to the fracturing of the dominant coalition at the end of its lifespan, typically due to the president trying to transform the dominant party's vision, which the legislature wants no part of, as they've spent their entire careers building and enhancing that vision. We see that every time Trump threatens to impose tariffs that never get implemented, or threatens to withdraw troops from some pointless occupation that ends up with only more troops.
With both the White House and Congress being held by the same dominant party, shouldn't the appointments go off without a hitch? Not necessarily.
Unfortunately, Jimmy Carter never had a SCOTUS opening, so we'll never know exactly what would've happened. But given what a clusterfuck his term was, we can guess that it wouldn't have gone well. He campaigned against the New Deal and Great Society that his party founded, and as an evangelical conservative. So maybe he appoints a pro-life justice only five years after Roe v. Wade, and it gets scuttled by the liberal wing of his own party in Congress, forcing him to nominate a pro-choice justice instead. The many lapses in government funding ("shutdowns") during his term centered around public funding for abortion (via Medicaid), so it was certainly bound to come up during any SCOTUS confirmation hearings.
In 1930, at the end of the Progressive Republican era, Hoover did get one of his nominees rejected by his own party that controlled Congress, albeit by just one vote. John Parker was protested by labor groups for favoring "yellow dog contracts," whereby a worker had to agree not to join a union as part of his hiring contract. That stance would've cut clearly against the Progressive GOP's promise to favor both labor and business, rather than business over labor -- an especially damning stance during the Great Depression. Hoover had to nominate a more populist justice, and chose an attorney who had worked on the investigation of the Teapot Dome scandal, Owen Roberts, who sailed through -- as did Hoover's other two nominations.
In 1860-'61, at the end of the Jacksonian era and just before the Civil War, Buchanan failed to confirm a nominee for a vacancy that opened up in June of 1860, the year Lincoln would be elected. He nominated Jeremiah Black during the lame duck session of 1861, but the Senate agreed -- by one vote -- to take no action. The spot got filled by Lincoln instead. Buchanan had chosen a wishy-washy northern Democrat like himself, which even his own party controlling the Senate did not feel comfortable accepting. Lincoln's successful appointment was an outright abolitionist. Earlier in his term, Buchanan did successfully appoint a justice, although narrowly.
In 1828, at the end of the Jeffersonian era, John Quincy Adams saw his first (and only successful) appointment die after two years in office -- and only slightly more than a month ahead of the presidential election, where he lost to Jackson. During the lame duck session, he tried to appoint a replacement for his own previous pick, yet his own party who controlled Congress decided to postpone action, and it got filled by Jackson instead. That reflected the fragmentation of the Democratic-Republican party at the end of the era: the Jacksonian faction blocked John Crittenden, who was a proto-Whig and anti-Jacksonian, so that president-elect Jackson could have a pick of his own. That choice, John McLean, began as pro-Jackson but evolved away toward the Whig party anyway.
In 1800, at the end of the early Federalist era, John Adams faced the resignation of the chief justice after having just lost his bid for re-election to Jefferson. During the lame duck session, Adams decided to nominate the first chief justice, John Jay. The Senate, controlled by his party, did agree to that choice -- but Jay declined, choosing to retire from politics altogether. Adams did successfully nominate his Secretary of State John Marshall to that vacancy during the lame duck session, and Marshall did choose to serve, not wanting the president-elect Jefferson to fill it with an anti-Federalist. (Adams had earlier appointed two justices successfully.)
Adams is thus the only disjunctive president whose failed appointment was not due to his own party in Congress blocking him, but to the nominee himself declining, and the only disjunctive president whose successful follow-up nomination after the failure was not bending to the shifting political winds. Quite the opposite, Marshall was a thorn in the side of the Jeffersonians for their entire era of dominance.
I attribute Adams' unusual status in this regard to the fact that there was not a strong party system during his time (the First Party System began with Jefferson), so he's not the best example of a disjunctive president. The founder of his era, George Washington, was a Federalist but ran and served as a non-partisan figure in order to calm tensions during the infancy of the nation.
At any rate, we see that Trump, like other disjunctive presidents, will face some kind of difficulty in nominating at least one of his would-be justices. He could pick someone a little too old-school for this phase of trying to transform the system, and have to go with someone who is more palatable to the evolving populist re-alignment.
It's the job of the Democrats and any not-so-corporate Republicans to make sure the populist principles he campaigned on are respected in his nominations to the Court. With previous disjunctive presidents, it was often only by one vote that the dominant party in the Senate decided not to go with someone so wedded to the old vision. There's no reason that can't happen again, especially comparing Trump to Hoover -- it could be over the same issue of labor unions for a supposedly progressive Republican, as an overly long economic expansion comes crashing down on the working class.