She then became visibly annoyed when he questioned the relevance of her question.
“It is [relevant] because he said ‘originally’ from,” she said, after telling him her own ancestors came from “Ireland and Italy.”
“You know everything [Mr Trump] said since. He’s put out a lot of tweets. He made himself available to all of you yesterday.”
Ms Conway, Mr Trump's usually unflappable aide, then snapped at reporters that the president is “tired.”
“A lot of people are sick and tired of this country - of America coming last,” she said, jutting her head towards the reporter whose ethnicity she questioned, “to people who swore an oath of office.”
She continued, saying people were “sick and tired” of “our military being denigrated,” and of the Customs and Border Patrol agents, who she claimed are “overwhelmingly Hispanic, by the way” being “criticised” and “doxxed.”
The people doing the "doxxing," she said, are “a bunch of Hollywood D-listers who have nothing else to do but sit on their asses on Twitter all day.”
She then called CBP agents “brave men and women who are diving into the Rio Grande to save people who are drowning, who are taking other people’s babies into custody and diapering them and feeding them, and looking the other way when people are running across with drugs.”
The press conference was the latest episode in the Trump administration’s response to overwhelming criticism of Mr Trump's racist tweets from the weekend.