“I’ll think about it,” Mr. Trump said. “It’s not something that I would want to do. It’s certainly something I would have standing to do. It’s not a priority. Because I’m beating him by so much, anyway, what do I have to bring a suit for?”

During his moment onstage, Mr. Cruz did not denounce Mr. Trump directly and did not mention his name. He offered himself as a trustworthy alternative to candidates who tell conservatives what they want to hear.

“How many people have been burned by politicians?” Mr. Cruz, who spoke less than two hours before Mr. Trump addressed the same group, said. “It happens over and over and over again.”

In a lawyerly fashion, Mr. Cruz laid out the argument that his actions validate his promises, presenting a list of trust “metrics,” such as battling the Affordable Care Act and refusing to give up on the fight against same-sex marriage.

He also made an effort to contrast with Mr. Trump stylistically. Last week, when a microphone failed at one of his rallies, Mr. Trump cursed the provider of the device and said that he should not be paid for his shoddy work. When Mr. Cruz’s microphone fell silent on Saturday, he calmly asked for a replacement and playfully tapped his fingers against it to determine if it was working.

Despite the recent infighting between the two candidates, supporters of Mr. Cruz and Mr. Trump did not seem to be persuaded by questions about Canadian roots or flip-flops on positions of the past.