The issue was just one of many covered in the broad-ranging interview, which also touched on guns, GOP congressional leadership, so-called birther conspiracies and his mother's voting record.
Cruz, who over the past few weeks has specifically endorsed deportations after months of resisting, said in an interview that aired Sunday on "State of the Union" that the U.S. should catch those who came here illegally under normal law enforcement practices, not through round-ups of the estimated 11 million undocumented people living in the U.S.
"No, I don't intend to send jackboots to knock on your door and every door in America. That's not how we enforce the law for any crime," Cruz told Tapper as his campaign bus ambled across northern Iowa.
Trump has promised to authorize a "deportation force"
to remove those immigrants from the U.S., pledging to do it "humanely."
Earlier this week, BuzzFeed captured Cruz on video
arguing that his plan was stronger than Trump's because he would not allow those back into the U.S., while Trump's would.
But on the actual act of deportation, Cruz is taking a softer stance.
"We don't have any system that knocks on the doors of every person in America," Cruz told Tapper. "We also don't have people going door-to-door looking for murderers. We don't live in a police state. We do have law enforcement."
Cruz indicated that he would only deport those who are apprehended, such as those who commit crimes or are caught by prospective employers without having immigrated legally. Immigrations and Custom Enforcement or Border Patrol agents could also apprehend those along the border.
Cruz confronts "birther" issue: My mom was born in Delaware
Cruz distanced himself from his family's connections to Canada, saying that his family never voted there.
"The Internet has all sorts of fevered swamp theories, but the facts are simple," Cruz told Tapper. "My mom was born in Wilmington, Delaware. She was an American citizen by birth. She's been an American citizen all 81 years of her life. She's never been a citizen of any other place."
Cruz, who was born in Canada but is a naturally-born U.S. citizen, has had to aggressively push back on recent questions pushed from Donald Trump about his eligibility for the presidency. He renounced his dual Canadian citizenship in 2014
. Earlier on Friday, Cruz's campaign shared with Breitbart a copy of his mom's birth certificate showing her born in the United States.
Cruz on campaign attacks
Cruz said his reluctance to engage directly with Trump, while frequently taking on rival Marco Rubio, doesn't mean he's afraid of the businessman.
"Marco and I have had some disputes on policy ... so I'm glad to engage on the substance there," he said. "But from the beginning of this campaign whenever anyone has thrown rocks, has thrown insults, I have not engaged, I have not reciprocated, and I don't intend to."
Cruz added: "And that's the way I've treated Donald, that's the way I've treated every other candidate and it's the way I intend to approach it. And the reason, Jake, (is) I think most Americans they couldn't care less about a bunch of politicians bickering like schoolchildren."
Cruz would continue to enforce law preventing guns for domestic abusers
Cruz also said he did not believe those convicted of domestic abuse should be allowed to buy a firearm, and added that the White House's reluctance to use the phrase "radical Islamic terrorism" -- a decision Cruz ties to political correctness -- has directly caused a loss of life.
"Well it's existing law right now that if you have a domestic violence abuse, you're not eligible to buy a firearm, and that is existing law," he said. "And I mentioned my obligation as president is to enforce the law and that is an obligation that I take seriously."
Cruz remains critical of GOP leadership in Congress
Cruz, who has often worked at odds with his Republican Senate colleagues, told Tapper, "the American people are so frustrated with Republican leadership."
"The Republican leadership passed this massive omnibus bill, over a trillion dollars, thousands of pages negotiated in the dark of night," he said. " It funds all of Obamacare, all of executive amnesty. It funds all of Planned Parenthood. It funds this disastrous Iranian nuclear deal."
The Texas senator even offered grudging support for comments made by Democratic leadership about the GOP-led Congress' recent actions.
"Nancy Pelosi and Chuck Schumer and Harry Reid were exactly right. All of them said publicly Republican leadership just funded our entire agenda," he said.
Cruz: I will not be "an emperor"
Cruz also had perhaps a surprisingly literal pledge when asked about repealing Obamacare, a law he has vigorously fought to undermine and repeal.
"The president doesn't have the constitutional authority to end it on Day One," he said, noting that he doesn't promise voters he will eliminate the law on his first day in the Oval Office.
"The only way to end it is to repeal that statute ... we will repeal Obamacare, but unlike Obama, I don't intend as president to be - to use his word - an emperor," Cruz said.
Cruz on how conservatives will win in 2016
The Texas senator also reiterated his election theory that past GOP presidential nominees have failed because they sought to moderate their positions for the general electorate, rather than stand up for their conservative principles.
"There are a lot of Washington political consultants that argue every election cycle the way for Republicans to win is run to the mushy middle, run a candidate that's Democrat-lite, and every time we do it we get clobbered, he said.
"You know this is no longer a theory ... this has been beta-tested. The one thing that is abundantly clear is that if we nominate another candidate in the mold of a Bob Dole, or a John McCain, or a Mitt Romney -- and all three of those are good honorable decent men who love this country -- but what they did didn't work," he said.