EXETER, N.H. — According to Nikki Haley, bullies are best subdued by a counter kick — in heels. Achieving a new vision for the country requires the leadership of a “tough-as-nails woman.” And generational change starts with putting a “badass woman in the White House.”

In ways both overt and subtle, Ms. Haley, the former United Nations ambassador and South Carolina governor, is setting up her 2024 presidential bid as the latest test of the Republican Party’s attitudes about female leaders. No woman has ever won a state Republican presidential primary, let alone the party’s nomination — and Ms. Haley is the first one to mount a bid since former President Donald J. Trump, who regularly attacked women in extraordinarily graphic and vulgar terms, rose to the head of the party.

The early days of Ms. Haley’s campaign, which she announced on Tuesday, quickly illustrated the challenges facing Republican women. For decades, female leaders in both parties have struggled with what political scientists call the