"John sometimes, he's a good person, but he's a little bit given to this kind of bellicose rhetoric, which has a tendency to inflame rather than to defuse them," Bayh said on CBS' "Face the Nation."
Naming Bayh as his running mate would also run the risk of losing Bayh's Senate seat to a Republican, a potential threat to the slim Democratic margin in the upper chamber of Congress.
Dark Horse: That could be former Georgia Sen. Sam Nunn, who has been a close Obama adviser on foreign affairs, particularly Russia, Stephanopoulos said.
Besides giving the Democratic ticket the foreign affairs gravitas that it currently lacks, having Nunn on the ticket would give the Democrats a shot at winnng Georgia.
"That would be huge," Stephanopoulos said.
Nunn, who is socially conservative, would create friction with some of the party's activists, particularly gays and lesbians.
But the most intriguing longshot may be Obama's former rival, Sen. Hillary Clinton, D-N.Y.
"You could never say never," Stephanopoulos said. "If you give me 50-1, I'd take it."