The Republican Party was at the brink of civil war on Sunday as Donald J. Trump signaled he would retaliate against lawmakers who withdraw their support from his campaign, and senior party leaders privately acknowledged that they now feared losing control of both houses of Congress.

Even before Mr. Trump’s second debate against Hillary Clinton, the party faced an internal rift unseen in modern times. A wave of defections from Mr. Trump’s candidacy, prompted by the revelation of a recording that showed him bragging about sexual assault, was met with boastful defiance by the Republican presidential nominee.

On Twitter, Mr. Trump attacked the Republicans fleeing his campaign as “self-righteous hypocrites” and predicted their defeat at the ballot box. In a set of talking points sent to his supporters Sunday morning, Mr. Trump’s campaign urged them to attack turncoat Republicans as “more concerned with their political future than they are about the country.”

The pressure from Mr. Trump did not deter new expressions of resistance on Sunday: Gov. Bill Haslam of Tennessee, a former chairman of the Republican Governors Association, announced he would not vote for Mr. Trump. So did multiple members of Congress, including Representative Kay Granger of Texas, the lone woman in the state’s large Republican delegation.