Kevin Hagen/Getty Images
In an interview with USA Today, Gingrich said such a tweet "makes you wonder about whatever else he's doing."
"The president of the United States can't randomly tweet without having somebody check it out," Gingrich said, later adding, "It undermines much more than a single tweet."
The former House speaker said later that Trump should not give up tweeting, however.
"He should tweet, but he ought to have an editorial board in-between the first draft and sending it," he suggested. "But he shouldn't give it up, and he shouldn't give up being Donald Trump. He got elected being this unique, charismatic, entrepreneurial guy who breaks lots of rules. ... If he starts getting to be normal, he'll cease being Donald J. Trump."
Top election officials from three states targeted by Trump in his tweets - California, Virginia, and New Hampshire - called Trump's assertion that massive voter fraud occurred in each "unfounded" and "unsubstantiated."
Gingrich also described a recent meeting he held with Trump, saying the president-elect told him that "this is really a bigger job than I thought."
"Which is good," Gingrich said. "He should think that ... you have war and peace, you have enormous powers ... and it all comes down to the Oval Office and it all comes down to you."