Gardner pointed to New Hampshire's role in the founding of the nation, saying that "we hold the first-in-the-nation primary, and we have a proud tradition of civic participation and responsibility."
"New Hampshire people aren't accustomed to walking away or stepping down from their civic duty. And I will not, either," Gardner said.
Gardner is the nation's longest-serving secretary of state and has been re-elected by Democratic and Republican legislatures, largely due to his advocacy for keeping the state's primary first in the presidential nominating process.
But he has faced sharp criticism for participating and hosting a meeting of the group created after Trump made a series of claims -- without ever offering any evidence -- that widespread voter fraud tipped 2016's popular vote and the outcome in New Hampshire in Hillary Clinton's favor.
A Democratic National Committee spokeswoman said Tuesday that Gardner "must step down from this sham of a commission."
"This commission is nothing more than an excuse to suppress the vote and discourage participation in our democratic process," said DNC spokeswoman Sabrina Singh. "Secretary Gardner should rescind any association he has with the fraudulent commission immediately."
New Hampshire's two Democratic senators, Jeanne Shaheen and Maggie Hassan, also called on Gardner to cut ties with the panel.
The commission's meeting is being led by Kris Kobach, the Kansas secretary of state who angered Democrats with a Breitbart.com op-ed
making the largely debunked claim that thousands of ineligible voters using out-of-state driver's licenses participated in New Hampshire's 2016 election and tipped the outcome in Clinton's favor.
Critics have pointed out that most of those voters were students attending college in New Hampshire who are legally allowed to vote in the state while living there -- even if they haven't updated their driver's licenses. They could also include active-duty military personnel stationed in New Hampshire and long-time residents who don't have or haven't updated their licenses.
"Granite Staters are not gullible or naive, and we do not appreciate those who impugn the integrity of our state's voting systems based on unsubstantiated accusations," Shaheen said in a written statement to the panel Tuesday.
"I am deeply concerned that falsehoods about illegal voting are being spread as a pretext for restricting access to the ballot box," Shaheen said. "This risks disenfranchising eligible voters and undermining faith in our democracy."