(CNN) Donald Trump on Wednesday and Thursday repeatedly called President Barack Obama the "founder" of ISIS and labeled Hillary Clinton the "co-founder" -- after which Clinton used his comments to say they show he is not fit to be president.

At a rally in Fort Lauderdale, Florida, Trump also said that ISIS "honors" Obama -- who the GOP nominee referred to as "Barack Hussein Obama."

After lamenting the "mistake" the US made by going to war in Iraq -- a conflict he repeatedly says that he opposed, despite evidence to the contrary -- Trump then criticized Obama's attempts to "clean up."

"Normally you want to clean up; he made a bigger mess out of it. He made such a mess. And then you had Hillary with Libya, so sad," Trump said.

"In fact, in many respects, you know they honor president Obama. ISIS is honoring President Obama. He is the founder of ISIS. He's the founder of ISIS, OK? He's the founder. He founded ISIS."

He then said that the "co-founder" of ISIS was Clinton.

Clinton on Thursday hit back in a series of tweets, saying it spoke to Trump's inability to lead the US.

"It can be difficult to muster outrage as frequently as Donald Trump should cause it, but his smear against President Obama requires it," Clinton tweeted. "No, Barack Obama is not the founder of ISIS. ... Anyone willing to sink so low, so often should never be allowed to serve as our Commander-in-Chief."

Anyone willing to sink so low, so often should never be allowed to serve as our Commander-in-Chief. — Hillary Clinton (@HillaryClinton) August 11, 2016

Clinton campaign adviser Jake Sullivan on Thursday accused Trump of "echoing the talks of (Russian President Vladimir) Putin."

"This is another example of Donald Trump trash-talking the United States," Sullivan said in a statement. "It goes without saying that this is a false claim from a presidential candidate with an aversion to the truth and an unprecedented lack of knowledge."

In an interview and campaign event in Florida on Thursday, Trump doubled down on his comments.

"Oh boy, is ISIS hoping for her," Trump said during an appearance before home builders in Miami Beach, portraying Clinton as lazy. "ISIS will hand her the most valuable player award. Her only competition is Barack Obama."

"He was the founder of ISIS, absolutely," Trump said of Obama earlier on CNBC. "The way he removed our troops -- you shouldn't have gone in. I was against the war in Iraq. Totally against it."

"That mistake was made. It was a horrible mistake -- one of the worst mistakes in the history of our country. We destabilized the Middle East and we've been paying the price for it for years. He was the founder -- absolutely, the founder. In fact, in sports they have awards, he gets the most valuable player award. Him and Hillary. I mean she gets it, too. I gave them co-founder if you really looked at the speech."

Did Obama underplay ISIS' influence?

JUST WATCHED Disgraced ex-congressman sits behind Trump at rally Replay More Videos ... MUST WATCH Disgraced ex-congressman sits behind Trump at rally 01:28

Critics say Obama has played down the terror threat and, while still on a victory lap after the killing of Osama bin Laden, failed to anticipate the rise of ISIS.

Trump later called Obama by his full name, "Barack Hussein Obama," in a jab at his handling of the Russian involvement in Crimea.

Conservatives opponents of the Obama administration have used the president's middle name to highlight what they see as Muslim sympathies and Trump had for a long time claimed Obama was not born in the US.

In fact, Obama is a Christian who was born in the US.

Following June's horrific nightclub shooting in Orlando, Trump had suggested that Obama had an ulterior motive.

"We're led by a man who is very -- look, we're led by a man that either is not tough, not smart, or he's got something else in mind. And the something else in mind, you know, people can't believe it," Trump said on Fox News' "Fox and Friends."

"People cannot believe, they cannot believe that President Obama is acting the way he acts and he can't even mention the words 'radical Islamic terrorism.' There's something going on. It's inconceivable. There's something going on."