"Anyone who thinks you're making a loyalty oath to some candidate fundamentally misunderstands this process," Cruz told reporters during a stop in Raleigh, North Carolina, one week before voters head to the polls there. "Listen, no political candidate is gonna make America great again. It's not gonna be Donald Trump, it's not gonna be me, it's not gonna be any candidate. The only power strong enough to restore this nation is we the people -- it is not about us, it's about the people."
Cruz added that the new practice at Trump rallies looks like people pledging their "loyalty to him, like subjects to a king."
Trump first asked supporters to make a pledge
at a rally in Orlando, Florida, over the weekend, jokingly asking them to raise their right hands and "solemnly swear" to vote for him.
Trump has sharpened his attacks on Cruz as the Texas senator has recently beat him in several states and has emerged as his top Republican challenger. Trump sharply questioned the senator's faith in a CNN town hall last month, saying "he holds up the Bible and then he lies."
"You can usually tell how dismayed he is by his volume, by the level of insults, and typically, when he attacks someone's faith, it is a sign that Donald is really worried," Cruz told CNN's Chris Frates at the gaggle on Tuesday. "I understand the last election day, Super Saturday, was a very bad day for Donald. He came in proudly expecting to sweep all four contests. Instead, he got clobbered. He got clobbered on election day."
Cruz also said that 2012 GOP nominee Mitt Romney, who has led the push against Trump winning the Republican nomination and recently cut robo-calls for Marco Rubio and John Kasich
, has offered to cut calls for the Texas senator.
"We have been in conversations with Gov. Romney. He's graciously reached out to our campaign and I've talked to him on the phone and we have been trading emails with him and I'm grateful for the support we are receiving all over the country," Cruz said.
Cruz, however, did not explain why Romney has yet to cut a call for his campaign.