Basic scenario that would keep the same 2 party system as IRL:
1. John W. Davis defeats Calvin Coolidge in 1924 and is reelected in 1928. He governs basically as Wilson's 3rd and 4th terms. Political activism for Republicans builds in enfranchised black communities.
2. Depression happens as IRL, 1932 is a landslide for a libertarianish New England Republican.
3. Republican incumbent wins with >60% in 1936, takes every state but MS/AL/GA/SC
4. Civil Rights Act of 1937 starts process but interpreted narrowly by Davis/Wilson controlled SCOTUS, military integrated in 1937-38.
5. 1932 Republican and his VP have appointed majority of justices by 1940, replaced entire SCOTUS by mid 1940's.
6. WWII against highly racist opponents with integrated military adds patriotic element to desegregation. Stricter laws passed during 1940-45 and fully enforced by new SCOTUS which has explicitly overruled Slaughter-House.
7. Congress passes federal law that new schools must be integrated during 1937-38. Enforcement lags and Democratic congress elected toward end of WWII tries to shut it down, but SCOTUS finishes the job in 1948 by ordering immediate compliance subject to contempt proceedings for county officials.
8. Dixiecrats run independent in 1948/52, leading the next Democratic president to tell them off and at least moderately embrace existing laws/SCOTUS rulings.
Republicans would be the clear socially liberal party to the present day. I'm not sure how things would shake out on economics, but one can imagine the increasing minority share of the Republican base in this world pulling them to the economic left over time, even though they would start out quite libertarian by modern standards. Jim Crow would still have been in existence more than long enough to be a defining political experience for the black community, so I think it would still have valence to the present day. For more dramatic changes, I like to imagine what might have happened if Benjamin Harrison had gotten the VRA-equivalent Force Bill through the Senate in 1890 and stopped the Southern Redeemer governments in their infancy.
This sounds like a dream scenario for someone like me. (paging Oldiesfreak!) I don't think it would necessarily lead to the Republicans being a socially liberal party. I can easily see how it would lead to a much, much softer stance on gun control than today, for instance, but I think it would reinforce the parochial and community-oriented tendencies found in parts of the Old Right.
The Republican Party would probably have some of its strongest supporters in the black middle class, which would likely be larger than it is today, but also have a dedicated base in rural southern blacks who bore the brunt of Southern oppression. All else equal, poor inner-city blacks would probably still lean towards the Democratic Party. For instance, a "law-and-order" approach would probably have appeal for black Republicans seeking to "protect their communities from crime" and, maybe somewhat paternalistically, "stem the spread of drugs and criminality in black American society", but the messaging would have to be considerably less dog-whistling and may not have been as effective for white voers as a result.
The resultant party would still be firmly on the right both in terms of economic and social policy, but it be less receptive towards the sort of hyper-individualistic, doctrinaire limited-government, and broadly "libertarian" currents that underlie most mainstream Republican thinking today. It would probably be less dogmatic-- social issues would almost always be framed in terms of in terms of their impact on society, communities, and social cohesion, rather than individual "liberty". It would have embraced the sort of policies laid out in the Moynihan Report targeted to improve the condition of black Americans, while still criticising welfare and the excesses of liberal permissiveness.
Black Republicanism would likely have Booker T. Washington as its intellectual godfather and his vision of black self-improvement and middle class advancement as its ideological core. It would still be broadly supportive of free-market, classical liberal policies, but ultimately they would drive the GOP in a considerably more "compassionate conservative" direction. At the same time, I could see Kirkian traditionalism, with its intense focus on the family, community, religion, and the value of tradition and custom (as opposed to progressive social engineering) having an appeal for certain segments of black Republicanism. It is quite interesting to think about, really.
I've completely ignored Southern whites, though. Frankly, they can only drag down whatever party they latch on to. If they stick with the Democrats, that party will likely fall into naked economic populism while taking a far more muddled approach towards social policy. It would firmly be the "anti-establishment" party, the standard-bearer of the white working class. Politics would be considerably more class-oriented. You'd have far fewer wealthy liberals backing them, although unionism would probably far stronger than it is today. On the flipside we'd be considerably to the right of where we are today in terms of social issues. Many blacks would probably oppose abortion as harmful to the black family, for instance, and quite a few would probably view it as a liberal social engineering project, akin to today's fringe view of abortion as "black genocide". Skill and Chances' map is pretty close to what I'd expect in this scenario, although I can't see Ike losing in 1952.
If southern Whites somehow latched onto the Republican Party on this scenario, on the other hand, it would produce a result not dissimilar from what we see today, only even more stark and probably sooner, as the black (and probably Hispanic) middle class would be considerably stronger and more influential than it is now. As such, their alienation from this GOP would have a greater effect.
