"The fact that Donald is now afraid to appear on a debate stage, that he doesn't want his record questioned ... that reflects a lack of respect for the men and women of Iowa," Cruz said.

Cruz noted that Trump has called him stupid in recent days, and said that should bode well for Trump in a debate between the two men.

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"He should do very, very well in a debate with someone who is so stupid away from moderators you’re so afraid of," Cruz said.

Cruz said that Trump is "scared" of moderator Megyn Kelly, with whom he sparred during a debate last year and later said she had "blood coming out of her wherever" during the debate.

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"Donald is a fragile soul. You know if she asks him mean questions I mean his hair might stand on end," Cruz said of Kelly in Fairfield, Iowa, and likened missing the debate to skipping a job interview.