SEOUL, South Korea — A popular South Korean singer, Lee Seung-chul, recently posted on Twitter what he called a sad joke that reflected public outrage over a scandal involving the country’s president, Park Geun-hye.

“If Hillary is elected, the United States will have its first female president. If Trump is elected, it will have its first crazy president,” went the joke, which was widely shared online. “South Korea got both in 2012.”

Ms. Park’s 2012 victory was hailed as a milestone for South Korea’s deeply patriarchal society. But four years later, pressure is mounting across the country and even from within her party for Ms. Park to step down or face impeachment. This week, she became South Korea’s first sitting president to be accused by prosecutors of a criminal conspiracy.

The scandal surrounding Ms. Park has left many South Korean women infuriated with the president and fearful that it could be used to argue that women are unfit to lead. They worry that the country, already among the lowest in global gender-equality rankings, could become even more resistant to elevating women to positions of power.