The two leaders were seen discussing the test and viewing documents in the public dining room of Mar-a-lago, the President's Florida resort, on Saturday night when the test was conducted.
"Accounts and photographs from other diners seem to indicate these communications occurred in the presence of other guests," Chaffetz wrote in a letter to White House Chief of Staff Reince Priebus.
CNN first reported that members of the upscale private club spotted top White House officials using flashlights from their cell phones to review documents at the table where Trump and Abe were having dinner.
White House press secretary Sean Spicer maintained that no classified information was exposed and that the President and other senior aides involved with the situation received all information on the North Korea missile launch in a secure area set up to safeguard sensitive materials.
House Speaker Paul Ryan also downplayed the incident on Tuesday, telling reporters, "It's my understanding that no classified information was discussed. And talking about foreign policy at the dinner table is perfectly appropriate."
But Chaffetz wrote, "discussions with foreign leaders regarding international missile tests, and documents used to support those discussions, are presumptively sensitive."
The Utah Republican asked for details on what security protocols were put in place at Mar-a-Lago, what documents were present at the dinner table and in "common areas" of the resort, and whether any officials were using cell phones while viewing any sensitive materials.
Chaffetz also asked Priebus to detail what kind of vetting procedures were put in place for guests or employees of the club 'in order to ensure that they are not foreign agents or spies on behalf of a foreign government."
While Chaffetz is raising concerns, House Intelligence Chairman Devin Nunes told reporters, "I don't know what the issue would be" and said he discussed the matter briefly with Chaffetz on the House floor. Nunes pointed to the administration's explanation and, asked whether a congressional inquiry was needed, said, "I don't think there is anything there."
The committee has not scheduled a hearing on the matter but Chaffetz gave a February 28 deadline for the White House to respond.
The probe comes as House Democrats are complaining that Chaffetz has been reluctant to use his committee to look into Trump's potential business conflicts. Democrats on the oversight panel also urged the GOP chairman to look into communications between now former national security adviser Michael Flynn and the Russian ambassador about sanctions that the Obama administration imposed in December.
"If House Republicans held President Trump to a fraction of the standard they held Secretary (Hillary) Clinton and requested documents and held hearings, that would be significant," Rep. Elijah Cummings, the top Democrat on the committee, told CNN in a written statement. "I hope the chairman will join our efforts to hold the Trump administration accountable to the American people. None of us have the right to remain silent."