Senator Tim Kaine said on Sunday that it was “not a coincidence” that the White House did not mention Jews or Judaism on Holocaust Remembrance Day yet Donald Trump signed an executive order banning travel from seven Muslim-majority countries.

“The final solution was about the slaughter of Jews,” said Kaine, Hillary Clinton’s running mate in her defeat by Trump in November, in an interview with NBC’s Meet the Press on Sunday. “We have to remember this. This is what Holocaust denial is.

“It’s either to deny that it happened, or many Holocaust deniers acknowledge, ‘Oh, yeah, people were killed. But it was a lot of innocent people. Jews weren’t targeted.’ The fact that they did that and imposed this religious test against Muslims in the executive orders on the same day – this is not a coincidence.”

Kaine spoke after White House chief of staff Reince Priebus, appearing on the same show, stood by the original statement.

“I don’t regret the words,” Priebus said, adding: “I mean, everyone’s suffering in the Holocaust, including, obviously, all of the Jewish people.”

On Friday, International Holocaust Remembrance Day, the White House said: “It is with a heavy heart and somber mind that we remember and honor the victims, survivors, heroes of the Holocaust. It is impossible to fully fathom the depravity and horror inflicted on innocent people by Nazi terror.”

Pressed on the omission on Saturday, after criticism from the Anti-Defamation League and Anne Frank Center, Trump spokeswoman Hope Hicks told CNN: “Despite what the media reports, we are an incredibly inclusive group and we took into account all of those who suffered.”

She added: “It was our honor to issue a statement in remembrance of this important day.”

Six million Jews were killed in the Holocaust, and during its ascent and the second world war the Nazi regime singled out Jewish people for persecution. Hicks also sent CNN a Huffington Post story about the millions of people who died for their ethnicity, sexual orientation, or political or religious beliefs.

Past presidents have marked Holocaust Remembrance Day with explicit mention of Judaism and antisemitism. In a speech on Friday, Israeli ambassador to the US Ron Dermer said it was “not only dangerous but even immoral” to remove Jews from the history of the Holocaust.

Kaine also linked the Trump administration’s break with precedent to Trump counselor Steve Bannon, the former publisher of Breitbart, a news site adopted by the so-called “alt-right”.

“I think all of these things are happening together,” Kaine said, “when you have the chief political adviser in the White House, Steve Bannon, who is connected with a news organisation that traffics in white supremacy and antisemitism, and they put out a Holocaust statement that omits any mention of Jews.”

In July, before Bannon’s arrival as campaign chief executive, Trump tweeted an image of Clinton superimposed over a pile of banknotes and a six-pointed star. He denied the image was antisemitic, saying the symbol was “a sheriff’s star” and blaming the media for the controversy, but also deleted the original tweet and replaced the star with a circle.

In August, Bannon denied a claim by his ex-wife that he made antisemitic remarks regarding his children’s school. In November, Senator Al Franken of Minnesota accused the Trump campaign of putting out an ad that used antisemitic tropes.

Fred Brown, a spokesman for the Republican Jewish Coalition, said the group believes Trump “holds in his heart the memory of the six million victims of the Holocaust, and is committed not just to their memory, but ensuring it never happens again”.

“The lack of a direct statement about the suffering of the Jewish people during the Holocaust was an unfortunate omission,” Brown said. “History unambiguously shows the purpose of the Nazi’s final solution was the extermination of the Jews of Europe. We hope, going forward, he conveys those feelings when speaking about the Holocaust.”

Trump’s use of the slogan “America First”, on the trail and in his inaugural address, has also attracted criticism from Jewish groups, including the Anti-Defamation League, which has noted the phrase’s origins with an isolationist and antisemitic 1940s movement that sought to avoid conflict with Nazi Germany.

Trump’s son-in-law, White House adviser Jared Kushner, is Jewish and his wife, Ivanka Trump, converted to Judaism when they married.

The executive order halting the admission of refugees and banning travel from seven Muslim-majority countries included an expression of intent to favour in the future religious minorities from those countries.

In an interview with the Christian Broadcasting Network broadcast on Saturday, Trump said he would prioritise Christians.

“It’s a religious test,” Kaine said on Sunday. “It imposes a different burden on Muslims than others. And the irony is not lost on me that it was issued the same day as the White House issued their Holocaust Remembrance Day proclamation that, unlike any previous administration, removed all reference to Jews.

“So you put a religious test on Muslims and you try to scrub reference to Jews in the Holocaust remembrance. This was horribly, horribly mishandled.”