WASHINGTON—Former Vice President Joe Biden said Sunday he didn’t believe he had ever acted inappropriately toward women but would “listen respectfully” to those who said he had, seeking to address a burgeoning controversy as he weighs entering the 2020 Democratic presidential primary.

“In my many years on the campaign trail and in public life, I have offered countless handshakes, hugs, expressions of affection, support and comfort,” Mr. Biden said in a statement. “And not once—never—did I believe I acted inappropriately. If...