Jarrod Rodriguez, a junior and the treasurer for the University of Florida College Republicans, said the impeachment move was “a mirror image of the partisan politics at the national level.”

He said he had attended the talk and never viewed it as a campaign event. The drive to oust Mr. Murphy, he said, showed that the opposition party would look for anything to remove the elected president.

“I think that they have no cigar — in both cases,” he said.

Theresa Chapman, a senior who was handing out fliers and pins for Planned Parenthood on campus, saw things differently. She said the funds Mr. Murphy had spent to bring the president’s son to town could have been better spent on something more beneficial and less divisive.

“Clearly he’s just not fit,” Ms. Chapman said of Mr. Murphy, adding that the same applied to President Trump. “He’s not here to support the entire student body. Only a portion of it.”

The visit by the president’s son filled a campus auditorium on Oct. 10, but he was frequently interrupted and many protested outside, The Alligator reported. Before Mr. Trump’s speech with Ms. Guilfoyle, whom he is dating, some students had called for Mr. Murphy and several other student leaders to resign.

But the correspondence between Mr. Murphy and Caroline Wren, a Republican operative who is now a fund-raising consultant with a re-election committee for the president, was the last straw for the student representatives who filed the impeachment bill. They said the emails, which were obtained through a public records request by Mariana Castro, an alumna, and shared with the student newspaper, were proof that the speech was a campaign event.

“Impeachment became the only option,” said Ben Lima, a senior and one of the student representatives who sponsored the bill.