"If @TedCruz doesn't clean up his act, stop cheating, & doing negative ads, I have standing to sue him for not being a natural born citizen," he tweeted.
Trump has previously argued that if Cruz won the Republican nomination, Democrats would argue that the Canada-born Texas senator was ineligible for the presidency. Last month, Trump asked a rally if he should sue Cruz "just for fun"
over the issue.
"How can Ted Cruz be an Evangelical Christian when he lies so much and is so dishonest?" the Republican front-runner tweeted.
Trump also had called Cruz's honesty into question in a tweet Thursday night.
"Lying Cruz put out a statement, 'Trump & Rubio are w/Obama on gay marriage.' Cruz is the worst liar, crazy or very dishonest. Perhaps all 3?" Trump said.
"There is more than a little irony in Donald accusing anyone of being nasty given the amazing torrent of insults and obscenities and vulgarities that come out of his mouth," Cruz told reporters in Greenville, South Carolina. "Being attacked by Donald, it is always colorful. I will give him this: he's not boring."
Cruz communication director Rick Tyler said Trump was having a "Trumpertantrum," a line the Texas senator has used in recent weeks.
"He needs to go to the time-out chair and think about his choices. That's what you would do with any 3-year old having a Trumpertantrum," Tyler said.
Trump and Cruz are in a tight race in South Carolina heading into the state's primary later this month. Evangelicals are a key voting bloc in the Bible Belt state and Trump is leading with white evangelicals in the state, according to an NBC/WSJ/Marist poll last month
But to keep Cruz, a Southern Baptist, from faring well, Trump has increasingly portrayed Cruz as fundamentally dishonest.
Trump previously slammed his chief presidential rival as a "total liar" as Cruz and his allies have claimed that Trump favors government-run, universal health care.
"Look, Ted Cruz is a total liar. I am so against Obamacare. I've been saying it for two years in my speeches, I'm going to repeal and replace Obamacare," Trump said
earlier this month on ABC's "This Week." "I don't even know where he gets this."