The Labour leader said the proposed venture would ensure "real privacy and public control over the data" that has made the US-based social media giant one of the world's richest companies.
Mr Corbyn, giving his first major speech on the media, also attacked the dominance of "unaccountable billionaires" and floated the idea of allowing journalists elect their editors.
Dubbed the British Digital Corporation, or BDC, Mr Corbyn said it would act as a sister organisation to the BBC and could be funded by a tax on the tech giants.
He said the "ambitious" proposal would be an "expanded iPlayer" to rival Netflix and Amazon.
"A BDC could develop new technology for online decision making and audience-led commissioning of programmes and even a public social media platform with real privacy and public control over the data that is making Facebook and others so rich," he said at the Edinburgh TV Festival's Alternative MacTaggart Lecture.
"It could become the access point for public knowledge, information and content currently held in the BBC archives, the British Library and the British Museum.
"Imagine an expanded iPlayer giving universal access to licence fee payers for a product that could rival Netflix and Amazon. It would probably sell pretty well overseas as well."
The proposal comes amid a data scandal engulfing Facebook over reports Cambridge Analytica harvested personal information about millions of users beginning in 2014.
How to stop Facebook from revealing everything about you
Show all 9
Mr Corbyn also used the speech to criticise television broadcasters for allowing the front pages of papers to set the news agenda.
"Just because it's on the front page of The Sun or the Mail doesn't automatically make it news," he said, adding there was a "herd instinct that isn't healthy in our journalism".
Turning to Fleet Street "media barons", Mr Corbyn said one option would be to "give journalists the power to elect editors and have seats on boards for workers and consumers when a title or programme gets particularly large and influential".
The Labour leader earlier floated the idea of ending "government control" over the BBC and allowing the public to elect the corporation’s board members.
His plans would see tech giants taxed to supplement the current income the BBC gets from TV licences and to pay for local “public interest journalism”.
Join our commenting forum
Join thought-provoking conversations, follow other Independent readers and see their replies