NEW YORK — Senator Tim Kaine, Democrat of Virginia, on Sunday compared Donald Trump’s encouragement of Russia to hack Hillary Clinton’s e-mails to the Watergate scandal that led to President Richard Nixon’s resignation.

Kaine, Clinton’s running mate, made the remarks on ABC’s “This Week’’ as he was being questioned about Clinton’s use of a private e-mail server while she was secretary of state, an issue that has dogged her throughout her campaign.

Kaine used the moment to attack Trump’s refusal to release his income tax returns and to attack his comments this summer that Russia should make public anything it might have stolen from Clinton’s e-mails, remarks he made after the disclosure that hackers had taken e-mails from the Democratic National Committee.

“He has openly encouraged Russia to engage in cyberhacking to try to find more e-mails or materials, and we know that this cyberattack on the DNC was likely done by Russia,’’ Kaine said of Trump on Sunday.

“A president was impeached and had to resign over an attack on the DNC during a presidential election in 1972,’’ Kaine added, referring to the Watergate scandal.

“This is serious business. So contrast the Hillary situation, where the FBI said there’s no need for legal proceedings, with an attack that is being encouraged by Donald Trump on the DNC by Russia, similar to what led to the resignation of a president 30 years ago,’’ he said.

During a news conference in July, Trump said he hoped Russian intelligence services had successfully hacked Clinton’s e-mail.

“Russia, if you’re listening, I hope you’re able to find the 30,000 e-mails that are missing,’’ Trump said at the time. “I think you will probably be rewarded mightily by our press.’’

Kaine wasn’t the only candidate to invoke Nixon’s name in an attack.

In a taped interview with NBC’s “Meet the Press’’ that aired Sunday, Governor Mike Pence of Indiana, Trump’s running mate, called Clinton “the most dishonest candidate for president of the United States since Richard Nixon.’’

Yet Pence also opened the door for more criticism of Trump by vowing to release his income tax returns within the next week.

Trump has refused to release his own returns, but Pence said his running mate would eventually disclose them as well, although he did not say when that might happen. Last week, The Washington Post revealed that Trump had to pay a $2,500 penalty to the IRS because his charitable foundation had made an improper political donation.

On CNN’s “State of the Union,’’ Rudolph Giuliani, the former New York City mayor who has become one of Trump’s closest advisers, suggested that Trump had backed away in his speech Wednesday from his proposal to carry out mass deportations of people who are in the country illegally.

Giuliani said Trump’s speech “leaves a very big opening for what will happen with the people who remain here in the United States after the criminals are removed and after the border is secure.’’

“Donald Trump, as he expressed in one of his interviews recently, would find it very, very difficult to throw out a family that’s been here for, you know, 15 years and they have three children, two of whom are citizens,’’ Giuliani said. “That is not the kind of America he wants.’’