The gender gap was much the same in Britain up to the 1990s with women favouring the Conservatives and men Labour in an even split election.
The history is different in the US because the Republicans were historically more positive towards women's causes (incl. prohibition, which was popular with female voters). Whereas the Democrats were supported by the groups with the most patriarchal family structure. The US suffragettes almost all came from Republican backgrounds.
The Nineteenth Amendment that gave women the vote was passed by 200-19 among Republicans in the House, but only 102-69 among Democrats. In the Senate it was 36-8 among Republicans, but 20-17 among Democrats.
The groups that were the most amongst female suffrage were German Lutherans, Catholics and Southerners. All Democratic core voters. The northern WASPs (and Scandinavian Lutherans), which formed the basis of the Republicans, were much more supportive of legal gender equality.
It was the cultural divide between "pietists" and liturgical religion with the Republicans attracting the (non-Southern..) pietists (non-conformists, Presbyterians, Scandinavian Lutherans) and the Democrats having the "liturgicals" (Catholics, German Lutherans, Episcopalians).
Prohibition was a major gender divider. In 1928 Herbert Hoover became known as "the woman's candidate" and was endorsed by the National Woman's Party
The depression did not completely eradicate this pattern. The best estimate for 1932 is FDR won 63-35 among men, but only 57-41 among women (based on voters recollection in 1936). Eisenhower won the female vote based on women blaming the Democrats for the Korean War, inflation, and corruption in Washington. The GOP was seen as the more responsible and moral party.
It was second wave feminism (incl. abortion), the South going Republican and the GOP starting to becoming perceived as the more warlike party, that changed the pattern. Probably also support for the "Great Society", but that is less clear.