Speaking at a privacy symposium in Sydney, Australia, the judge said that legislation would protect privacy, but also safeguard freedom of expression on the internet.
Lord Leveson was criticised for only giving the internet a very limited mention in his 2,000-page report on UK press regulations and standards last month.
He was accused of being out of touch after the report suggested that online material did not have the same reach and credibility as traditional print media.
The judge recommended a new independent self-regulatory watchdog to oversee the press, backed by legislation.
Newspaper editors met on Thursday (December 6) to implement his "broad proposals" for self-regulation, although without new laws as the judge had recommended. They have promised to report back to the government "very shortly" on how to implement the Leveson plan.
Lord Leveson's inquiry into the press was launched following public revulsion that the News of the World had hacked into the voicemail of murdered schoolgirl Milly Dowler after she went missing in 2002.
But the backdrop of the inquiry was also the broader shift of news consumption from print newspapers to the internet, where people access particularly breaking stories on news websites, but also on blogs and social networks such as Twitter.
A recent high-profile case exposed the danger of unshackled speculation online, after a BBC Newsnight report claimed an unnamed senior Conservative had been involved in a child abuse scandal in north Wales, leading to Tory peer Lord McAlpine being wrongly linked with the abuse by Twitter users.
Whilst the BBC has apologised to Lord McAlpine and paid him compensation, Lord Justice Leveson said that those on the internet "were not so restrained".
He said that there was "a danger of trial by Twitter" and other forms of social media, and it would "take time to civilise the internet".
But the judge feels that new laws would most likely be required to maintain both privacy and freedom of expression in the digital age.