By Dan Merica, Eric Bradner and Theodore Schleifer CNN

(CNN) -- President Donald Trump called on Wednesday for "a major investigation" into voter fraud, following through with baseless claims he has made since November's election alleging millions of illegal votes during the general election without citing any evidence.

The probe into voter fraud may not be limited to the 2016 presidential election and could center on large states where Trump didn't necessarily compete, according to White House press secretary Sean Spicer, who added, without citing evidence, that problems with illegal voting occurred in bigger states.

"I will be asking for a major investigation into VOTER FRAUD, including those registered to vote in two states, those who are illegal and ... even, those registered to vote who are dead (and many for a long time). Depending on results, we will strengthen up voting procedures!" Trump wrote in two consecutive tweets.

Trump's comments on voter fraud came Monday during a meeting with congressional leaders, where he reiterated an unsubstantiated claim that 3 to 5 million illegal votes cost him the popular vote, according to two sources familiar with the meeting.

On Tuesday, Spicer vigorously defended Trump's statement about illegal voters, though neither Trump nor his surrogates could provide evidence that any substantial illegal voting had occurred or influence the popular vote.

"I think in terms of registration, where you have got folks on rolls that have been deceased or moved or registered in two counties -- this isn't just about the 2016 election. This is about the integrity of our voting system," Spicer said Wednesday.

Trump faced widespread criticism for his remarks, including from some congressional leaders in his own party, and Democrats have alleged that Republican efforts in the name of fighting voter fraud has the effect of preventing or delaying legal voters who traditionally back Democratic candidates

Ohio's secretary of state replied to Trump's tweet on Wednesday, saying his office already investigated claims of voter fraud.

"I responded ... for Ohioans, to let them know we already have an investigation, or review as we call it, underway," Jon Husted, a Republican, told CNN's Carol Costello Wednesday. "In the past ... we found less than a thousand cases of voter irregularity, less than a couple hundred cases of voter fraud, and zero cases of voter suppression."

Considering Ohio is home to about 7 million registered voters, these cases constitute "a small number," Husted said.

The reviews, which take place every two years, are best conducted at the state level, Husted said.

"I don't think that federal involvement is important in this particular matter, because the states run the elections," Husted said. "We don't want federal involvement in our elections, we want to keep this in the hands of the states -- that's where it should be."

Former Democratic National Committee chair Rep. Debbie-Wasserman Schultz issued a harsh condemnation of Trump's call for an investigation, telling CNN's "New Day" that the tweet's message was "deeply disturbing."

"He seems to be questioning the legitimacy of his own election, all while, for the last couple of months, touting how legitimate and huge his election was and historic it is. It can't be both," she said.

"What is the most deeply disturbing about his penchant for lying is -- if he is willing to lie about the trivial, like crowd size, or the significant, like voter fraud, then what happens if -- God forbid -- we go to war and we have our troops lives on the line and there are causalities? Is he going to send Sean Spicer out to lie about the causalities that have taken place? Are our allies going to be able to trust us?"

And former President Barack Obama's senior adviser David Axelrod responded on CNN on Wednesday by saying, "It becomes an impetus for those who want to further erode voter protection for people who legitimately want to vote and are facing a series of barriers."

When pushed by the media about whether Trump will call for an investigation into the matter, Spicer said, "maybe we will," adding later that Trump believes in voter fraud based on "studies and information he has."

"The President does believe that, I think he's stated that before, and stated his concern of voter fraud and people voting illegally during the campaign and continues to maintain that belief based on studies and evidence people have brought to him," Spicer said at Wednesday's press briefing.

Trump lost the popular vote to Hillary Clinton by nearly 3 million votes in November, but won the Electoral College and thus the presidency. Trump, however, has seemingly been fixated on the popular vote, mentioning voter fraud in regards to his popular vote loss a number of times since November.

Spicer did not say specifically Tuesday which studies the President was using to support his claim of 3 million to 5 million illegally cast votes, but a 2014 study by Jesse Richman and David Earnest found more than 14% of non-citizens in 2008 and 2010 "indicated that they were registered to vote." Only US citizens can vote in federal elections.

However, Brian Schaffner, a political scientist at the University of Massachusetts, told CNN on Tuesday that his study that is apparently being cited by the White House was misinterpreted and did not support the administration's claims.

A number of studies have also found no evidence of widespread voter fraud.

The Truth About Voter Fraud, a report written by experts at The Brennan Center for Justice, found voter fraud rates were between 0.00004% and 0.0009%.

Trump himself -- through his lawyers -- has also argued that there was no evidence of voter fraud in the 2016 election. In a court filing objecting to Green Party candidate Jill Stein's Michigan recount petition, lawyers for the president wrote, "All available evidence suggests that the 2016 general election was not tainted by fraud or mistake."

Legal issues on voter rolls

Judges have identified examples of systemic efforts to distort the voter rolls in recent years.

But those efforts hurt traditionally Democratic voters and help Republicans.

In North Carolina, a federal appeals court wrote in 2016 that -- after receiving data on the use of voting practices by race -- the Republican-controlled state legislature enacted a series of laws designed to suppress African-American turnout with "almost surgical precision."

The state required voters to present forms of identification disproportionately used by white people. It cut back on early voting, eliminated same-day voter registration and preregistration for voters under 18, and it eliminated Sunday voting -- with the state even arguing in court that "counties with Sunday voting in 2014 were disproportionately black" and "disproportionately Democratic."

US Circuit Judge Diana Motz wrote in July 2016, in a ruling striking down North Carolina's, that the laws were "enacted with racially discriminatory intent."

"The General Assembly enacted them in the immediate aftermath of unprecedented African American voter participation in a state with a troubled racial history and racially polarized voting," Motz wrote.

Still, many North Carolina counties worked around Motz's ruling. Guilford, the state's third-largest county, opened just one polling place for the first week of early voting. Charlotte's Mecklenburg County chopped its early voting locations from 22 when early voting began in 2012 to 10 in 2016, with voters reporting waiting in line for three hours.

Democrats also believe that new voter ID laws played a role in lowering turnout in other states -- particularly Wisconsin, where Hillary Clinton narrowly lost to Trump. Turnout there dropped nearly four points from 2012, despite record early voting numbers.

CNN's Eleanor Mueller contributed to this report.

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