Mrs Trump's appearance came as the billionaire attempted to repair serious damage to his reputation among female voters.
He stumbled badly over the last week after suggesting women having illegal abortions should face criminal "punishment".
His campaign manager was also charged with assaulting a female journalist and there was an angry backlash when he posted an unflattering image of Mrs Cruz on Twitter.
An unusually humble Mr Trump said: "Was this my best week? I guess not. I could have done without the tweet. Yeah it was a mistake. If I had to do it again, I wouldn't have sent it."
On his controversial abortion comments, Mr Trump said: "That was a hypothetical question. As a hypothetical question I would have rather answered it in a different manner, yeah."
Mr Trump's first wife Ivana defended him, saying she speaks to him regularly and tells him to "be calm".
She told the New York Post: "Donald cannot be calm. He's very outspoken. He just says it as it is."
She added: "I don't think he's feminist, he loves women but not a feminist."
In a lengthy interview with The Washington Post, Mr Trump predicted that the United States was on the verge of a "very massive recession" and claimed he could wipe out the country's $19 trillion debt in eight years, which many economists disputed was possible.
Mr Trump said: "I think we're sitting on an economic bubble, a financial bubble. I'm talking about a bubble where you go into a very massive recession."
He said it was a "terrible time right now" to invest in the stock market.
Such comments are usually avoided by presidential candidates but Mr Trump said he didn't care if he upset Wall Street, adding: "I'm the Lone Ranger."
The businessman said in his first 100 days in office he would renegotiate trade and military deals, including the US role in Nato.
He claimed renegotiated trade deals would lead to economic growth and allow the US to pay off its debt.
However, opponents said he would end up starting a devastating trade war with China.
Mr Trump also said he would use an "aura of personality" to increase the US standing in the world.
He admitted that "I do bring rage out" and said it was necessary to "break some eggs" during the campaign.
But the billionaire said, if he won, he would become "so presidential" and the "most boring human being ever".
