Herman Cain says he'd like to be secretary of Defense

Herman Cain

is no longer running for president. He doesn't think he's likely to be picked as the eventual nominee's running mate, either. But he does have a Cabinet post in mind should the

GOP

win the

White House

next fall: secretary of Defense.

Cain shared the idea with ABC's

Barbara Walters

during her annual special on the "10 Most Fascinating People." And the veteran journalist was just a bit surprised. (see video below)

"What?!" she asked. "Not Treasury?"

Cain said he thought he'd be a good pick because he wants to "influence rebuilding our military the way it should be."

Walters, still stunned, reminded Cain of his struggles during the campaign with topics like Libya. And there is always Uzbeki-beki-beki-stan-stan.

"I have been doing my homework ever since that difficulty," Cain said.

In a way, Defense seems like a good fit for Cain. He's still playing defense even as a non-candidate over allegations that surfaced just as his campaign peaked about marital infidelity and sexual harassment.

"I believe that in the court of public opinion I have not been treated fairly. Because the accusations were false, but they continued to be spun as if they were true," he told Walters.

Did he think, she asked, that his race played a role?

"Yes, because I happen to believe that the

Democrats

did not want an accomplished, articulate, optimistic black man to face President Obama," he said. "I can't prove that, Barbara, but I do believe that."

Though the charges were untrue, he maintained, he had to drop his campaign because of the pain they caused his family. "They were not going to stop," he said.

He said his wife is "at peace" with him, and did not ask him to drop out.