"I saw that Speaker (Paul) Ryan said some things tonight about getting to the bottom line," the Ohio Republican said at a CNN town hall with Vermont independent Sen. Bernie Sanders. "Frankly, I think he should be more aggressive. I think he should speak out more and hopefully he will."
A spokesperson for Ryan released a statement earlier Tuesday.
"We need to have all the facts, and it is appropriate for the House Oversight Committee to request this memo," AshLee Strong said.
Kasich, who ran against Trump in the Republican presidential primary last year, said "this is not a time for Republicans to hide" but also argued it's not a "time for Democrats to exploit."
Kasich said he was open to the idea of a select special committee but wasn't ready to join Democrats in their call for a special prosecutor.
"The things that have swirled around this White House are the reasons that caused me not to move forward and support him both in the primary and going to that Republican convention," he told CNN's Dana Bash and Jake Tapper, who moderated the discussion.
The two 2016 presidential candidates sat down for a CNN town hall Tuesday night, just hours after the New York Times first reported that Comey wrote in a memo
in February that Trump asked him to end the investigation of Flynn. CNN has not viewed the memo but sources described it to CNN.
Sanders likened such a request from the President to the definition of "obstruction of justice" and reiterated his calls for a special prosecutor to get involved in the probe of alleged ties between Trump's campaign and Russia.
Kasich was a vocal critic of Trump throughout the campaign last year and declined to attend the GOP convention that took place in his home state of Ohio.
After speaking to Trump on February 14, Comey was so appalled by the request that he wanted to document it, sources told CNN. Comey shared it with FBI senior officials, according to sources.
"I hope you can let this go," Comey wrote, quoting the President.
Kasich argued that he tried to raise alarms about Trump during the campaign last year.
"Now that he's in, it's like rooting for the pilot on the airplane that I'm on, but sometimes I think we have to get into the cockpit and tell the pilot 'no we need to do something different,'" he said, adding that Trump needs to "understand the gravity of the situation."
Kasich and Sanders got into a heated argument over whether they considered Trump a liar or not. Sanders conceded that there is usually a learning curve for a president and mistakes will be made, but argued "there is something unusual about this president" and questioned whether he had "an authoritarian-type mind."
"I don't think this is just a learning curve, I think he's a smart guy, but something else is going on," Sanders said.
When Kasich wasn't willing to go as far into speculating Trump's intentions, Sanders asked him directly: "Is Trump a liar?"
"Well, you know, sometimes he says things I don't agree with and I think they don't resemble the facts," Kasich said.
But, he continued, he's not going to "call somebody a liar" and start using names. "Let the facts speak for themselves and then we can draw a conclusion."
Still, Kasich argued that the current scandal is "a long way" from the same conclusion that resulted from the Watergate scandal with President Richard Nixon.
"Ultimately the Republicans said there was something wrong that was done, they went down to the White House and said 'Mr. President you have to resign.' But we're a long way away from anything like that."
Both men found agreement on wanting to see the investigations move forward in a bipartisan way, and they also agreed on wanting to hear more dialogue about policy related issues on the national level.
So would the two be willing to run on the same ticket one day?
"I think John is thinking about becoming a democratic socialist," Sanders quipped in a reference to his own political identification.
"I don't have any doubt you're not becoming a serial, entrepreneurial Republican," Kasich said in return.
Asked more seriously about their political futures, Sanders declined to weigh in on whether he plans to run again in 2020 and criticized the media for looking ahead to the next election.
For his part, Kasich said he wasn't sure but wanted to "keep a voice."
"I don't know where the world is going to take me. I don't know what the Lord's purpose is for me. But I'm going to try to figure it out. It may not be politics, and it may be. Who knows?"