Mr. Cornyn sought to discourage members of his own party from challenging the electoral results further when Congress meets in a joint session in early January to tally and finalize the outcome.

“That would be a bad mistake,” he said. “I think there comes a time when you have to realize that despite your best efforts, you’ve been unsuccessful. It’s sort of the nature of these elections. You’ve got to have a winner and you’ve got to have a loser.”

Senator Charles E. Grassley of Iowa, the longest-serving Republican, captured the grudging acceptance of many of his colleagues after the Electoral College action when asked by a reporter if he now considered Mr. Biden the president-elect.

“I don’t have to — the Constitution does,” he said as he hustled to a vote. “I follow the Constitution.”