Senators Jeff Flake and Susan Collins were among the lawmakers who took issue with comments Trump made regarding Christine Blasey Ford, a university professor from California who detailed her sexual assault allegation against Kavanaugh at an extraordinary Senate Judiciary Committee hearing last week.
At a rally later in Southaven, Mississippi, Mr Trump mocked last week's Senate testimony by Prof Christine Blasey Ford, without mentioning her by name.
"What neighborhood was it in? I don't know", Trump said, flailing his arms around to mock her as he has some of his political foes or journalists during similar events in the past. I don't know. I don't know. I don't know. But I had one beer.
The president warned about a unsafe and "damn sad situation" in the country where a man's life can be destroyed over accusations over misconduct and are assumed guilty instead of innocent. To discuss something this sensitive at a political rally is just not right.
"It's just not right". Flake added, "I wish he hadn't done it".
Speaking to reporters at the White House on Tuesday, Mr Trump said: "My whole life I've heard, "you're innocent until proven guilty", but now you're guilty until proven innocent".
In a Los Angeles Times story - one of many that have described the president's love of the show, and the co-hosts' reciprocal fondness for the president - Kilmeade argued that he has taken Trump to task many times.
But the three swing Senators were joined by other Republican Senators who are firmly supporting Kavanaugh in criticizing the remarks.
In an interview with Today on Wednesday morning, Republican Sen.
Those GOP senators, along with Democrats Heidi Heitkamp of North Dakota and Joe Manchin of West Virginia, have yet to declare how they will vote on Kavanaugh. "The president is very confident in his nominee, as he's stated time and time again, and we expect the Senate to vote, and we hope they do that soon", she said. Since then, two other women, Debrah Ramirez and Julie Swetnick have come forward with additional allegations, although Swetnick's does not involve Kavanaugh directly.
In a Twitter post on Wednesday, Trump did not back away from his comments at the rally, instead attacking Kavanaugh's Democratic critics.
Isn't there something "wrong" with the President of the United States mocking somebody who says that she was sexually assaulted? wondered CNN's Jim Acosta.
Dudley has spoken out in support of Kavanaugh's nomination, and on Monday the White House released joint statements from him and a former Yale suite mate, Dan Murphy, rebutting any claims that Kavanaugh drank to the point of blacking out.
White House counsellor Kellyanne Conway also defended the president.
Ford says Kavanaugh tried to rape her at a 1982 party. There are facts that she can not remember: "how she got home, how she got there, where the house was, what the date was".
The fight over Kavanaugh's nomination comes against the backdrop of the #MeToo movement fighting sexual harassment and assault that has toppled a succession of powerful men. Voters will cast ballots in November 6 elections in which Democrats are trying to seize control of Congress from Trump's party.