“The legislature is going to put a bill on the desk of the next governor to ban abortion,’’ said Josh Shapiro, a Democrat running unopposed for the party’s nomination for governor. “Every one of my opponents would sign it into law, and I would veto it.”

Mr. Shapiro, the state’s attorney general, has been primarily known for defeating multiple cases brought by supporters of Donald J. Trump claiming fraud after he lost Pennsylvania by 80,000 votes in 2020. When Mr. Shaprio began his campaign last year, he focused on voting rights, but he said in an interview last week that he expected the general election to become a referendum on abortion.

His campaign said it had its best day of fund-raising after the Supreme Court draft leaked last week.

He rejected the notion that voters, whose attention spans can be short, will absorb a major Supreme Court reversal and move on by the fall. “I’m going to be talking about rights — from voting rights to reproductive rights — until the polls close at 8 p.m. on Election Day,’’ Mr. Shapiro said. “People are very concerned about this. I expect that level of concern, of fear, of worry, of anger is going to continue.”