An apology from Ted Cruz Rafael (Ted) Edward CruzSenate Republicans face tough decision on replacing Ginsburg Cruz: Trump should nominate a Supreme Court justice next week Renewed focus on Trump's Supreme Court list after Ginsburg's death MORE for calling Majority Leader Mitch McConnell Addison (Mitch) Mitchell McConnellObama calls on Senate not to fill Ginsburg's vacancy until after election Planned Parenthood: 'The fate of our rights' depends on Ginsburg replacement Progressive group to spend M in ad campaign on Supreme Court vacancy MORE a liar "ain’t gonna happen," the presidential hopeful said during a Thursday interview with CNN's Dana Bash.

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"If the Washington lobbyists want to see that happen, they can hold their breath a long, long time," Cruz said. "My focus is on the American people."

Last summer, Cruz made headlines when he accused McConnell of lying to him over a deal to vote on reauthorizing the Export-Import Bank.

He brushed aside suggestions from Bash that an apology could help unite more Republicans behind him in his race for the party's nomination.

"This is why people are so frustrated with Washington," Cruz said. "It's the inside battles back and forth. This isn't a game. This isn't about Washington power brokers. This isn't a smoke-filled room."