"They have to go," the real estate mogul who is leading Republican primary polls nationally and in the key early-voting states of Iowa and New Hampshire, said in an interview aired Sunday on NBC's "Meet the Press."
In the wide-ranging interview, Trump also declined to rule out shutting down the federal government over Planned Parenthood.
He called for the United States to seize oil in Iraq -- particularly the oilfields controlled by ISIS. And he warned of a "nuclear holocaust" if the Iran deal is implemented.
But Trump's top campaign issue so far has been his focus on immigration. He told NBC Obama's moves that allow undocumented immigrants who were brought into the United States as children to stay, as well as actions that allow the undocumented parents of U.S. citizen children to avoid deportation.
Trump said he would deport families "together." They'd be required to leave the United States and only then could some come back under an expedited process, he said, because "we either have a country or we don't have a country."
He also said he opposes "birthright citizenship" -- or a right to citizenship for any child born within a country's borders, regardless of their parents' legal status.
"You have to. What they're doing -- they're having a baby," Trump said.
He didn't reprise his attacks on Obama's citizenship -- which shot Trump into the political spotlight four years ago -- but wouldn't answer when asked whether he believes the President was born in the United States.
"I don't like talking about it anymore," he said. "Because honestly I have my own feelings."
Trump also said he isn't sure whether he has ever donated to Planned Parenthood -- which he attacked for offering abortions in some of its clinics.
"I give to so many, Chuck. I give to so many organizations over the years," Trump said. "Hundreds of millions of dollars. And so I don't know. I don't think so. But it's possible somewhere."
But he demurred when asked whether he would shut down the federal government over Medicaid dollars going to pay for women's health services at Planned Parenthood -- which many Republican presidential contenders have advocated.
"It's something I'd have to think about, to be honest with you. I don't want to give a hard and fast answer to that," Trump said. "It bothers me greatly that they're doing the abortions. At the same time, women's health issues are, you know, very important to me."
On foreign policy, Trump advocated seizing Iraqi oilfields that are controlled by ISIS in order to "take away their wealth."
"You go and knock the hell out of the oil, take back the oil," he said. "We take over the oil, which we should have done in the first place."
Pressed on the possibility that doing so might require ground troops, Trump said: "That's OK."