Alwyn Cashe rescued six troops from a burning vehicle during an enemy attack while his body was engulfed in flames, but current federal law renders him ineligible for the award because more than five years have elapsed since his actions.
"'Can I give it to myself anyway?'" the president continued, prompting laughter from the audience. "They said, 'I don't think that's a good idea.'"
"I want him to be remembered as the best father that anyone could ask for, and also at the same time being the best soldier that anyone could ask for," said the son of a soldier set to receive the Medal of Honor. "He was my icon."
A former U.S. Army special forces medic, who saved scores of his fellow service members and Afghan commandos during a fierce firefight in 2008 will receive the Medal of Honor.
Retired Captain Gary Rose was welcomed following a long period of denial by Pentagon officials about the full scope of operations in Laos.
Sergeant William Shemin and Private Henry Johnson were recognized "for conspicuous gallantry" nearly a century after serving.