Cruz and presidential rival Sen. Marco Rubio (Fla.) have led the Republican field in calling for the Senate to refuse to consider a nominee to replace Scalia this year. Senate Majority Leader Mitch McConnell (R-Ky.) seemed to agree Saturday, saying in a statement that the vacancy should "not be filled until we have a new president."
Cruz dismissed the notion that the Senate has an obligation to consider an Obama nominee since the presidential election is still about nine months away. "Not remotely," he said on NBC's "Meet the Press."
Scalia's unexpected death has upended the political debate and introduced a new dynamic into the 2016 Republican presidential primary, as candidates seek to convince voters that they should pick Scalia's replacement.
Rubio, speaking on CNN, praised Scalia. "When I'm president of the United States, I'll nominate someone like Justice Scalia," he said.
Still, not every member of the Republican presidential field is adamant that the Senate should not consider an Obama nominee this year.
Former Florida governor Jeb Bush suggested on CNN that the Senate ought to weigh a potential justice, but he predicted that Obama's pick would be "out of the mainstream" and ultimately rejected in a floor vote.
"The Senate has every right not to confirm that person," Bush said, adding that the Senate "should not confirm someone who is out of the mainstream."
In contrast with Cruz and Rubio, Bush said it was "really not important to me" whether the Senate decides to vote on an Obama nominee or refuses to consider one altogether.
"It's up to Mitch McConnell. It's really not important to me," he said.
With less than a week to go before the South Carolina Republican primary, Cruz and others are beginning to tailor their arguments about the court -- and related attacks on fellow White House hopefuls -- to South Carolina voters.
In his interview, Cruz pressed the issue of gun rights.
"Let me say something in particular to the veterans of the state of South Carolina, ... your Second Amendment rights are hanging in the balance," he said on ABC.
"If an additional liberal justice goes to the court, we're one justice away from the Second Amendment being written out, and if Donald Trump becomes president, the Second Amendment will be written out of the Constitution. ... Whether it's Hillary, Bernie or Donald Trump, the Second Amendment will go away."
Trump is currently leading the field in South Carolina with 35 percent, according to a RealClearPolitics polling average. Cruz is next with 16 percent, followed by Rubio with 14 percent.