Reviewing, and possibly eliminating, private contractors from the security clearance process is part of the department's ongoing review of the mass shootings at the Navy's Washington headquarters last Monday.
Alexis, who was honorably discharged from the Navy. gained access to the facility due to his status as a civilian contractor.
"Bottom line, we need to know how ... warning flags were missed, ignored or not addressed in a timely manner," Carter told reporters at the Pentagon.
Defense officials say Alexis obtained a security clearance while he was in the Navy and was not subjected to a second background check when he became a contractor after leaving the service in 2011.
Earlier this month, a Defense Department official told reporters that Alexis was not subjected to a reinvestigation for his security clearance after being hired for the IT job at the Navy Yard.
Reinvestigations for security clearances could be triggered when a former military member is transitioning into contractor or civilian service, according to Pentagon officials. But that only occurs if the time between retirement and re-entry is more than two years and if “derogatory information” is uncovered, officials say.
Concerns over how Alexis and other civilian workers receive security clearances reached a fever pitch on Capitol Hill in the aftermath of the Navy Yard massacre.
Sen. Dianne Feinstein (D-Calif.), head of the Senate Intelligence Committee, said last Wednesday the panel planned to hold hearings on the security clearance process.
"It may be time for a [congressional] review to see how well these contractors are doing their jobs" in terms of vetting candidates for sensitive, national security positions, Sen. Jeff Sessions (R-Ala.) said.
Sen. Lindsey Graham (R-S.C.) openly questioned whether the process government agencies use to vet civilians before allowing them access to highly sensitive national security issues is "fundamentally broken."
"What are we doing to check who works for the government in a civilian capacity?" asked Graham, who is a senior member of the Senate Armed Services Committee. "Is it because we do not have the resources, or is the system just fundamentally broken?"
That said, "the lion's share" of the security clearance work done for the U.S. government, including the Defense Department, is carried out by the Office of Personnel Management (OPM), according to a Pentagon official.
While the majority of security clearance checks are conducted by government employees, a fair share of those investigations for military or national security positions are handled "by a mix of government employees and [private] contractors," the official said.
While it remains unclear whether mistakes made in contractor-run clearance investigations led to Alexis being granted access to the secure Navy facility, Carter noted the issue was not strictly a Pentagon problem.
"It is bigger than our department," he said, adding a White House-led inquiry into the clearance process would have the final say in whether private contractors should be cut out.
That administration review will be conducted by officials from OPM, the Office of Management and Budget and the Office of the Director of National Intelligence.
Recommendations from that review are expected to be completed no later than December, according to Carter.