He said a House vote to strip Greene of her committees would set a new standard regarding what members of Congress say before they're elected and for the majority party intervening in the committee assignments of the minority.
By a vote of 145-61, Republicans voted not to strip Representative Liz Cheney from her position as the No. 3 House Republican.
The House will vote today on whether to remove Greene from her committee assignments.
Republican Senator Todd Young of IN said this week Greene was an "embarrassment" and "there's no place for her IN our party".
"My interpretation of what happened with McConnell is that he very much would like to turn the page and move on from Trump and left the door open quite intentionally to see if there was the requisite amount of support to get rid of him", says Brendan Buck, a Republican consultant who once advised former House Speaker Paul Ryan.
Greene has faced bipartisan criticism, including from Senate Republican Leader Mitch McConnell, who warned against "loony lies and conspiracy theories" that he said "are a cancer for the Republican Party and our country".
In a sign of how consuming the problem has become Mr McCarthy met with Ms Greene on Tuesday and Wednesday. "I'm fine with it and I can live with that", he said.
"It's me this week, and it'll be someone else next week, and our leaders are too weak to stand up against it".
In repulsing the leadership challenge, Cheney has passed her first political test, but she still has much fix work to do in Wyoming, where voters overwhelmingly disagree with her impeachment vote and she already has primary challengers.
House Republicans voted on Wednesday to keep Rep. Liz Cheney in GOP leadership despite her vote to impeach former President Donald Trump last month, with a strong majority of the conference rejecting the challenge from Trump allies to punish the Wyoming Republican.
Every attempt at reaching some sort of resolution to avoid a floor vote by the whole chamber - an apology by Mrs Greene, sanctions imposed by the Republican leadership or some compromise acceptable to Democrats - has been in vain.
The party's dilemma, we're told, is captured by two members of Congress.
The Democrats "have no business determining which Republicans sit on committees", Ms. Cheney said Thursday.
House Minority Leader Kevin McCarthy released a statement Wednesday night condemning Greene's past comments but didn't indicate that any party disciplinary action would be taken against her. It also would have served as a warning shot to traditional conservative Republicans hoping to diminish Trump's hold over the party.
The freshman Georgia lawmaker cannot be ousted from her congressional seat, but the Democrats could unite to deny her committee assignments.
She burst onto the national political scene after just a month in office and with enthusiastic support from Trump.
"They're going to judge her on things that were said that she has now denounced, before she was ever a member of Congress", House Republican leader Kevin McCarthy said. A group of House GOP lawmakers introduced a measure that would leave Mrs. Greene on her committees but remove Rep. Ilhan Omar (D., Minn.) from the House Foreign Affairs Committee. King was defeated in a Republican primary election last June.
The Senate plans to begin its impeachment trial of Trump next week.
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