© ReutersTommy Robinson arrives for his hearing at the Old Bailey
The EDL founder was removed from Facebook and Instagram on Tuesday for violating their policies on "organised hate" and the targetting of Muslims.
He responded by posting videos to Youtube - the only major social media platform he can still use - claiming he was the victim of censorship.
In a letter to Google chief executive Sundar Pichai on Friday, Mr Watson described Robinson - real name Stephen Yaxley Lennon - as a "violent, racist, Islamophobic campaigner".
He wrote: "I am now writing to implore you as a matter of utmost urgency to follow the lead that has been, belatedly, set by Facebook, and remove forthwith all 'Tommy Robinson' and related pages from your YouTube platform."
Mr Watson said that social media platforms need to be accountable and not "avoid their social responsibilities".
He added: "I have recently been making the argument that the social media companies have failed to regulate hate speech and harm on their platforms.
"I would ask that you immediately close down all of Yaxley-Lennon's sites on YouTube before the virus of his views grooms countless more followers via your platform."
Mr Watson said that Robinson had gained more than 30,000 new subscribers to his video channel since he was banned on Facebook and Instagram.
Youtube has previously suspended adverts on Robinson's videos but is coming under increasing pressure to remove them altogether. He has already been banned by Paypal and Twitter.
On Wednesday Damian Collins, chairman of the Digital, Culture, Media and Sport Committee, joined calls for Robinson to be banned from Youtube.
Mr Collins said: "His YouTube channel has hundreds of thousands of followers and includes films viewed by millions of people.
"Far-right groups are exploiting social media to spread their messages of hate, and the YouTube 'next up' feature helps them by directing viewers to even more of this content once, as soon as they start to engage with it."
YouTube operates a hate speech policy which removes content promoting violence or hatred against individuals or groups based on attributes including religion, race, immigration status or nationality.
It uses a "three strikes" system, giving offending users a warning and removing the specific video. Channels are only terminated if policies are repeatedly violated.
Following his removal from Facebook, Robinson insisted he would continue his campaigning even if YouTube deleted his account.
"You thought deleting us from Twitter would stop us - it didn't," he said during a 16-minute video.
"You thought removing our PayPal would stop us. It affected us, they stopped our donations by 70 per cent. Did it stop us? No it didn't. If you think deleting us from Facebook or Instagram will stop us... no it won't."
Reacting to the move on Tuesday, Robinson claimed the "corrupt media and the establishment" were trying "to silence any opposition to their globalist plans".
He said: "I've breached no laws of Facebook, everyone is going to know that I've breached no rules, what I've done is shown people the truth and that is what they are removing, the truth. People will still find me."
Comment: Social media giants in league with politicians are increasingly claiming themselves to be the arbiters of free speech, and this nefarious drift towards censorship is alarming because it's being used to shut down much more legitimate outlets and voices than just Tommy Robinson's: