For months, Mrs. Clinton had warned that the stakes were high in this election because of the Supreme Court seats the next president would most likely have to fill. But Justice Scalia’s death at a West Texas ranch on Saturday, and the instant fight over whether Mr. Obama should name his successor, turned an abstract stump-speech line into an urgent and overriding argument, as candidates in both parties hurried to respond to the seismic shift the vacancy created.

Officials said they hoped the vacancy would prompt a sobering moment, causing voters to reconsider their romances with Senator Bernie Sanders of Vermont and Mr. Trump. For Democrats, it will drive home how instrumental the court has been on health care, same-sex marriage, gun control, voting rights and abortion. For Republicans, who fills the vacancy will determine whether Mr. Obama’s legacy can be rolled back or is cemented.

What is beyond doubt is that the battle will push the parties farther apart, ensuring that an already rancorous election year becomes even more contentious as Republicans and Democrats alike try to rally their voters by portraying the opposition’s approach to the nomination as illegitimate.

The future of the court dominated the presidential contest on Sunday, with notable differences among the Republicans in how they sought to exploit the vacancy, as each candidate sought to gain the most benefit heading into Saturday’s primary in heavily conservative South Carolina.

Jeb Bush’s campaign said the vacancy underscored his contention that he is the only candidate prepared for the serious issues the next president will face, and questioned whether Mr. Trump could be relied on to appoint a conservative jurist. Gov. John Kasich of Ohio, portraying himself as a voice of reason who could bridge the partisan divide, said he hoped Mr. Obama would refrain from nominating someone but stopped short of urging the Senate to refuse to consider an Obama nominee. And Mr. Rubio, vying for conservative support in South Carolina, said Mr. Obama could “nominate whoever he wants” but added: “We’re not moving forward on it. Period.”