Cruz said Trump's campaign sent a cease-and-desist letter to his campaign Tuesday stating that an advertisement Cruz's campaign aired about Trump is "replete with outright lies, false defamatory and destructive statements" and that the campaign must take it down immediately or face a lawsuit.
The ad, titled "Supreme Liberty," states that the nation is one Supreme Court justice away from losing "life, marriage, liberty, the Second Amendment." It then cuts to a 1999 clip of Trump on "Meet the Press," stating he is "very pro-choice" and would not ban partial birth abortion.
"We cannot trust Donald Trump with these serious decisions," the ad states.
Cruz said the cease-and-desist letter is "one of the most remarkable letters I've ever read" and goaded Trump.
"I have to say to Mr. Trump, you have been threatening frivolous lawsuits for your entire life. Even in the annals of frivolous lawsuits, this takes the cake," Cruz said. In a response letter to Trump's lawyer, Cruz's counsel said the ad would now be aired with greater frequency.
"So, Donald, I would encourage you, if you want to file a lawsuit challenging this ad claiming it is defamation, file the lawsuit," Cruz said, stating it would be thrown out of court, and Trump's lawyers would be reprimanded for filing a frivolous suit.
"I look forward to any lawsuit. One of the things I look forward to most of all is deposing Donald Trump," Cruz said. "For that particular endeavor, I may well not use outside counsel. I may take the deposition myself."
In a statement, Trump said he is against abortion and for the Second Amendment and knocked Cruz for having to apologize to Ben Carson after the Iowa caucuses. Trump has threatened to sue Cruz before, over whether Cruz is eligible to be president because he was born in Canada.
"If I want to bring a lawsuit it would be legitimate. Likewise, if I want to bring the lawsuit regarding Senator Cruz being a natural born Canadian I will do so. Time will tell, Teddy," Trump said.
Trump said Cruz is a "liar" and that his ads and statements are "clearly desperate moves by a guy who is tanking in the polls — watching his campaign go up in flames finally explains Cruz’s logo," Trump said.
Standing behind a white table, Cruz spent about 25 minutes unloading on Trump and Sen. Marco Rubio, with whom Cruz is fighting for second place in polls here in South Carolina. Cruz rattled off a laundry list of issues with the two men and why he thinks they should not be the nominee.
Like most lawyers, Cruz relied on visual aids, unveiling a poster of Democrats to whom Trump has given campaign contributions that sat on an easel. It listed 30 Democrats and Democratic organizations, including that Trump gave $9,500 to Democratic presidential candidate Hillary Clinton, $84,000 to New York Gov. Andrew Cuomo and $21,000 to former New York governor Eliot Spitzer, whose name, along with some others on the list, was misspelled.
"Donald Trump is a direct cause of Obamacare passing," Cruz said. "Donald helped fund the Democratic takeover of both houses of Congress."
Cruz repeated a warning that Trump would appoint liberal justices to the Supreme Court in the wake of the death of Justice Antonin Scalia, pointing to Trump's past statement that his sister, a federal appeals court judge in New Jersey, would make a good justice.
Trump's sister, Maryanne Trump Barry, wrote the majority opinion in a 2000 case declaring New Jersey's ban on partial birth abortion unconstitutional. Cruz said she is a "radical pro-abortion federal judge.”
Cruz said he wants Trump to answer questions about whether he wants the federal government to fund Planned Parenthood, which Trump said does "wonderful things" at a Republican debate Wednesday, and federal judges under oath.
"Whether in a deposition or a court of law, getting Donald Trump under oath, under penalty of perjury, answering these questions, well, I’ll point out it didn’t work very well for Bill Clinton," Cruz said.
"Donald Trump does not want to be under oath answering questions about his own record because his position quite simply is that anyone who points to his record is somehow lying," Cruz said.
Within barely a minute of taking the stage at the Sun City retirement community in Bluffton, S.C., on Wednesday morning, Trump was attacking Cruz.
Trump said that in every speech he mentions his support for the Second Amendment and getting rid of Obamacare — only to have Cruz then accuse him of the opposite.
"I mean, they lie. And the reason that I can say that nicely is that Rubio called him a liar, also, so once another guy called him a liar, it's easier for me to call him one. But really: Cruz does lie. I've never seen anything quite like it," Trump said.
Cruz also uncorked on Rubio (R-Fla.), accusing the campaign of falsely accusing the Cruz campaign of creating a fake Facebook post from Rep. Trey Gowdy (R-S.C.), one of Rubio's top surrogates here.
"It’s unfortunate. His campaign has not gotten the traction he wanted," Cruz said of Rubio.
Cruz said Rubio is "behaving like Donald Trump with a smile."
Cruz has long said that he does not, and will not, engage in insults and wants to keep the race focused on substance; he said he has still done that.
"I’m not insulting anybody here," Cruz said. "I am responding with simple facts."
The Texas Republican has said that he plans to make 2016 a referendum on the Supreme Court. Cruz said he would block the confirmation of federal court judge Sri Srinivasan to the Supreme Court, someone rumored to be on President Obama's short list of potential nominees. Cruz voted to confirm Srinivasan to the federal appeals court of the District of Columbia in 2013 and has called him a "friend" but said that Obama should not appoint a justice in an election year.
"Voting for a candidate for the D.C. circuit is very different from confirming someone to the U.S. Supreme Court," Cruz said. "We should not confirm a justice nominated during an election year."
Jenna Johnson in Bluffton, S.C., contributed reporting.