Schofield, who was presenting This Morning on ITV, handed the Prime Minister a card with six names on it, saying he had found them after a brief trawl of the internet.
The camera angle meant at least four of the names were visible.
Downing Street called the action a "silly stunt" and the Prime Minister warned it could spark a witch-hunt against gay people.
Mr Cameron's spokesman later warned against people "throwing accusations around and trial by Twitter" after the names had been posted online.
Schofield had asked Mr Cameron : "You know the names on that piece of paper. Will you be speaking to those people?" .
Mr Cameron said: "I've heard all sorts of names being bandied around and what then tends to happen is, of course, everyone sits around and speculates about people.
"I do think it's very important that anyone who's got any information about any paedophile, no matter how high up in the country or whether they are alive or dead, go to the police."
But the Prime Minister added: "There is a danger, if we're not careful, that this can turn into a witch-hunt, particularly against people who are gay."
Schofield later said: "If any viewer was able to identify anyone listed, I would like to apologise and stress that was never my intention. I was not accusing anyone of anything. It is essential it is understood that I would never be part of any kind of witch-hunt."
Downing Street sources described the broadcaster's actions as "irresponsible", adding: "This silly stunt has resulted in people's names being put out there. They will want to vigorously defend themselves. "
This Morning has been reported to broadcasting regulator Ofcom by Tory MP Rob Wilson.
l Police investigating abuse allegations against Jummy Savile have arrested his former chauffeur, Ray Teret, plus a second man in Manchester.