Despite being a Democrat and a supporter of Wilson, if Cox somehow won the election, this does not necessarily guarantee an American entry into the League of Nations.
Besides the fact it might not get approved by the senate anyway, Cox only reluctantly supported Wilson's LoN. If he beats Harding, I don't think he would push the issue too hard and he would move onto other things. His first term, if he won again in '24, he would probably sign an act to assist those suffering from unemployment during the 1920/21 depression.
As for who he would nominate to replace Chief Justice White, maybe James F. Byrnes?
In my timeline I will probably go with someone like Byrnes, but it was known that Cox was a progressive of sorts. I have made some wikiboxes for Cox's cabinet, which is:
The Cabinet of President James M. Cox (D-OH)
Vice President: Franklin D. Roosevelt (D-NY)
Secretary of State: John F. Shafroth (D-CO)
Secretary of Treasury: J. Swager Sherley (D-KY)
Secretary of War: Henry L. Stimson (R-NY)
Attorney General: Charles Allen Culberson (D-TX)
Postmaster General: James Farley (D-NY)
Secretary of Navy: Theodore Roosevelt Jr. (R-NY)
Secretary of the Interior: Gilbert M. Hitchcock (D-NE)
Secretary of Agriculture: Henry Cantwell Wallace (FL-IA)
Secretary of Commerce: Herbert Hoover (R-IA)
Secretary of Labor: Rueben G. Soderstrom (I-IL)