'I would not vote for' Rep. Marjorie Taylor Greene: Gov. Asa Hutchinson

Republican Arkansas Gov. Asa Hutchinson said Sunday that he would not vote for freshman GOP Rep. Marjorie Taylor Greene, R-Ga., who has come under intense criticism for comments and actions she made before and during her time in office. But Hutchinson stopped short of saying she should be punished for her controversial comments.

"She's long embraced conspiracy theories like QAnon, voiced support for executing Nancy Pelosi. Is she fit to serve?" ABC's "This Week" Co-anchor Martha Raddatz asked Hutchinson.

"I would not vote for her. The second question is, should the House of Representatives make a disciplinary call on her? I'm not going to get in the middle of that. They're going to have to make that judgment," the governor said. "The people of her district elected her and that should mean a lot. They elected her and she's going to run for reelection and she'll be accountable for what she said and her actions."

CNN's KFile, in a report on her online activity, said Greene allegedly "liked" a comment on her Facebook page in 2019 that threatened the lives of House Speaker Nancy Pelosi and other Democrats. ABC News cannot confirm the "likes" because the posts have been deleted. While Greene hasn't disavowed the posts, she later tweeted that several people had managed her social media accounts.

"Over the years, I've had teams of people manage my pages," Greene wrote. "Many posts have been liked. Many posts have been shared. Some did not represent my views."

Pelosi ripped into Republican leadership Thursday for giving Greene a position on the House Education and Labor Committee despite the first-term congresswoman's support of conspiracy theories claiming mass school shootings at Sandy Hook and Parkland were staged.

"What could they be thinking? Or is thinking too generous of a word for what they might be doing," Pelosi told reporters. "It is absolutely appalling."

House Republican Minority Leader Kevin McCarthy, R-Calif., has also said he would have a "conversation" with Greene over her support for threatening social media posts and is reportedly planning to speak to the Georgia representative next week.

But Greene is a close ally of former president Donald Trump, and aided his efforts to overturn the results of the 2016 election. Greene tweeted Saturday that she had spoken with Trump and that he supported her.

Amid calls from some House Democrats to strip Greene of her committee post and potentially expel her from the chamber, Hutchinson told Raddatz, "I don't think we ought to punish people -- from a disciplinary standpoint or party standpoint -- because they think something a little bit different."

"Governor, you say you shouldn't go after someone because they think of something a little bit different. She believes in conspiracy theories, that there are pedophiles running Washington. That's not just a little bit different," Raddatz interjected.

"Whenever you have a broad diversity of the party reject the extreme elements, it's not mainstream GOP, and that's what we've got to get back to," Hutchinson replied. "We've got to have a regard for those people that supported Donald Trump … because they have a message. They have a concern. But at the same time, we don't want to gloss over the terrible actions that happened at the Capitol."

Controversy over Greene's comments and actions comes as the Republican Party ponders its political future after former President Donald Trump's administration.

In the aftermath of the Jan. 6 storming of the Capitol, Hutchinson blamed Trump for the riots and said "I want this administration to end" when asked if the president should resign. But Raddatz pressed Hutchinson about Trump's ongoing influence within the GOP.

"There's a lot of different voices (in the party) and the Republican leadership has said very clearly, including Kevin McCarthy, that President Trump has -- bears responsibility for … (bringing) people to the Capitol," Hutchinson said. "He brought them to Washington. They went to the Capitol. He bears some responsibility there."

"The Senate trial is going to refocus what happened on the attack on the Capitol and it's going to call all Republicans to take a position more clearly. President Trump has helped build the party in the last four years, I hope he does not help to destroy the party in the coming four years," the governor added.

As Washington prepares for the upcoming second impeachment of Trump lawmakers are also trying to balance managing the coronavirus pandemic as logistical problems hamper the countrywide vaccine roll out.

With reports of new COVID-19 strains reaching the United States, President Joe Biden has spent much of his first days in office laying out his administration's response to the pandemic -- including issuing an executive order to invoke the Defense Production Act to ramp up supplies needed for manufacturing and deploying the vaccine.

Hutchinson said that vaccine distribution has been "seamless" under the Biden administration.

"In terms of the vaccine distribution, it's been seamless. And I was delighted that we had a ... 14% increase in vaccine supply last week. This is going to be very, very important for us. (Biden) said they're going to invoke the Defense Production Act. I don't know the details on that, but anything they can do to speed up the production," the governor said.

"Thank goodness we have that partnership which is good with the federal government. And President Biden and his team is -- is working to assure that partnership and not tear it apart, which I'm very grateful for," he added.

Earlier on "This Week," Raddatz also spoke with top officials in West Virginia, a state that has been praised for its successful vaccination roll out as many of other states struggle.

Part of West Virginia's success is that the state opted out of a federal vaccination plan -- which partnered with national pharmacy chains -- and another program to vaccinate people in nursing homes and long-term care facilities. Instead, the state worked with a network of local pharmacies and long-term care facilities, partnering with the National Guard to distribute the vaccine.

Raddatz asked West Virginia Gov. Jim Justice what other states should do differently.

"We had to take the vaccines to the people, rather than bring the people to the vaccine," Justice said. "So there's just been a lot of work but it's just, it's just practical smarts, that's all there is to it."

"I do congratulate West Virginia," Hutchinson said in response. "They did a great job in their roll out. And I think the message is that every state has to have flexibility -- do something that works for their population. We've actually done many of the same things they did in West Virginia, which is utilizing our local pharmacies … (and) the National Guard."

Arkansas has administered less than 65% of its distributed vaccine doses while West Virginia had dispensed nearly 85%, according to The New York Times.

When pressed why Arkansas had not administered more vaccine, Hutchinson replied, "Ours is moving in that direction. We are working in our rural communities, as well as in our urban centers. And whenever you're at 60%, you're asking where's the 40% of the vaccines. When we receive them, the first dose is in people's arms within 72 hours. That's our goal. That's what we did last week."

ABC News' Mariam Khan contributed to this story.