Most Revd Justin Welby has also denounced racism against Jewish people, after figures last week showed anti-Semitism's reached record levels.
Speaking to the Times ahead of the launch of the All-Party Parliamentary Inquiry into anti-Semitism, he said: "If someone wrote a letter a letter to the Times expressing wildly, revoltingly anti-Semite views, would the Times feel it had to publish it?
"So why is it OK for Twitter and Facebook to publish stuff that is the expression of hatred on the basis of race and ethnicity and is in many cases clearly criminal?
"To put on Twitter a photograph of someone on a bar of soap because they're Jewish, that is so disgusting they shouldn't allow it to go up. They should remove it at once. And stop the account of anyone who is doing that. It would mean monitoring but there's the software to do it."
The All-Party Parliamentary Inquiry into anti-Semitism wants prosecutors to examine if prevention orders like those used to restrict sex offenders' internet access could be used. The cross-party group also highlighted the use of anti-Semitic terms online.
Last week, a Community Security Trust report said UK anti-Semitic incidents more than doubled to 1,168 in 2014.
The trust, which monitors anti-Semitism in Britain, says this was its highest figure recorded since it began work in 1984.
