"One of the ways I can fight back is to bring a lawsuit against him relative to the fact that he was born in Canada and therefore cannot be President," Donald Trump said in a statement. "If he doesn't take down his false ads and retract his lies, I will do so immediately. Additionally, the RNC should intervene and if they don't they are in default of their pledge to me. "
The billionaire businessman later held a press conference in which he said he "never ever met a person that lies more than Ted Cruz."
Trump continued to heap criticism on Cruz in an interview with CNN's Jim Sciutto Monday afternoon on "The Lead," saying that with "guys like Ted Cruz -- it's all talk and no action."
"He's a very bad guy. He's a very nasty guy. He's got no endorsements from any senators who he works with. He'll never be able to unify anything," Trump said of Cruz.
He continued: "He can stand in the middle of the Senate floor and let everybody laugh at him, which is what happened last time, as you know -- he got nothing out of it, zero -- but, you know, that's not the kind of person who you're going to need to turn the country around."
Trump acknowledged that his brawl with Cruz might harm the eventual Republican nominee, whoever it is, "but I think we'll get over it."
The Cruz campaign responded by saying Trump is angrily lashing out "with name-calling and falsehoods."
"To be fair to Donald, he doesn't know what he actually believes so he can't really be expected to defend himself with reason and facts," said Rick Tyler, Communications Director for the Cruz campaign. "He can't be held to account for his own words because he likely doesn't remember from one day to the next what he says about anything because he has no core set of beliefs."
The Trump-Cruz war is escalating ahead of South Carolina's Republican primary on Saturday. The two repeatedly clashed during a debate over the weekend and Cruz released an ad to supporters on Sunday highlighting the importance of the next president's role in selecting a replacement for the late Supreme Court Justice Antonin Scalia.
Cruz isn't backing town, telling reporters earlier Monday that Trump was aligning with Democrats after he used the debate to blame former President George W. Bush for trouble in the Middle East.
"When Trump sided with MoveOn.org and Michael Moore and the fever swamp left-wing, that demonstrated where he was coming from," Cruz said during a stop in Aiken, South Carolina.
Trump veered between his favorite topics at a rally in Mount Pleasant, South Carolina, earlier Monday, including his plan to build a wall along the Mexican border. But he repeatedly came back to his criticism of Cruz.
"I've been in a business where you know, it's pretty sharp. You meet sharp people, I don't mean sharp like sharp, although they're that also. But you meet people that really go to the edge," Trump said. "I've never met people like politicians, they are the most dishonest people I've ever met ... I think Jeb (Bush) is just Jeb. But this guy Ted Cruz is the most dishonest person I've ever met in politics."
Trump also complained that Cruz effectively stole the Iowa caucuses from him by having staff tell Ben Carson supporters that the retired neurosurgeon was dropping out of the race -- which was not true.
"He apologized to Carson after the event, he should have apologized to me," Trump said. "If Iowa had any guts, the people from the Republican Party, they should disqualify him from winning Iowa. I really mean it. Because what he did was a fraud."