In a press conference Wednesday, Trump said that the 30,000 missing emails from Clinton's private email server would reveal "some beauties."
He then issued a plea: "Russia, if you're listening, I hope you're able to find the 30,000 emails that are missing. I think you will probably be rewarded mightily by our press. Let's see if that happens. "
Trump later tweeted "If Russia or any other country or person has Hillary Clinton's 33,000 illegally deleted emails, perhaps they should share them with the FBI."
Clinton's campaign pounced on the remarks.
"This has to be the first time that a major presidential candidate has actively encouraged a foreign power to conduct espionage against his political opponent," said Hillary for America Senior Policy Advisor Jake Sullivan. "That's not hyperbole, those are just facts. This has gone from being a matter of curiosity, and a matter of politics, to being a national security issue."
CBS2's Dick Brennan reported the embarrassing revelations forced the resignation of DNC Chairwoman Debbie Wasserman Schultz.
Trump dismissed the claims saying it's not clear who hacked those emails, but the hacking is a sign that foreign countries no longer respect the U.S.
Trump's running mate Mike Pence said in a statement there should be "serious consequences'' if Russia is found to be interfering in the U.S. electoral process.
Pence added that "both parties and the United States government will ensure there are serious consequences'' to any hacking.
Trump said he has no relationship to Russian President Vladimir Putin and does not know if Russia or some other country, such as China, is responsible for the DNC breach.
Incoming DNC Chairwoman Donna Brazile said Trump's comments were appalling.
"I'm appalled at Mr. Trump encouraging a foreign power to interfere in the election, but more importantly, interfere the affairs of the Democratic National Committee," Brazile said.
Former CIA Director Leon Panetta echoed those sentiments.
"I find those kinds of statements to be totally outrageous, because you've got now a candidate who is in fact asking the Russians to engage in American politics," Panetta said. "And I just think that's beyond the pale."
Clinton's camp has suggested the Russian hack is a plot to benefit the Trump campaign.
"Further, experts are saying is that then, because they possessed those emails, that Russian state actors were feeding the email to hackers for the purpose of helping Donald Trump," Clinton campaign spokesman Robbie Mook said.
A Trump spokesman said the Republican nominee was not inviting Russia to do anything, only pointing out that Clinton's private email system was vulnerable to hacking.
Trump told CBS Miami Wednesday that he rejects allegations that the Russians are trying to help him get elected in November.
"I don't know anything about it. I can tell you, I think if I came up with that, they'd say, 'Oh it's a conspiracy theory,' it's ridiculous," Trump said. "I mean I have nothing to do with Russia. I don't have any jobs in Russia. I'm all over the world but not involved in Russia."
Trump's campaign manager Paul Manafort told "CBS This Morning" Wednesday Trump had no financial relationship with wealthy Russians.
"That's what he said. That's what... that's obviously what our position is," Manafort said. He added that Trump will not be releasing his tax returns, which could prove or disprove that claim, because he is under audit.
"It has nothing to do with Russia, it has nothing to do with any country other than the United States, and his normal tax auditing processes -- so that issue will be dealt with when the audits are done."
Trump has previously tweeted he has no investments in Russia.
Newt Gingrich took to Twitter to say that Trump's comment was a "joke," adding that Trump's call couldn't be considered a national security issue because the emails Clinton deleted were personal.
(TM and © Copyright 2016 CBS Radio Inc. and its relevant subsidiaries. CBS RADIO and EYE Logo TM and Copyright 2016 CBS Broadcasting Inc. Used under license. All Rights Reserved. This material may not be published, broadcast, rewritten, or redistributed. The Associated Press contributed to this report.)
for more features.