The incident occurred after the final whistle blew at the Tottenham Hotspur Stadium on January 15, which saw Arsenal win 2-0.
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Watts, 35, appeared at Uxbridge Magistrates' Court on Friday to confirm his name, age and address before pleading guilty to assault by beating.
He also admitted to charges of going onto an area adjacent to the playing area and throwing a missile onto a football playing area after throwing four coins onto the pitch.
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Having watched the north London derby from behind the goal, at the end of the match, Watts became involved in an altercation with Ramsdale.
After the England international retrieved his water bottle and towel from near the advertising boards, the Tottenham fan kicked the goalkeeper in the back.
Watts managed to flee the scene as stewards tried to get hold of him before covering his face and leaving the stadium.
However, footage discovered the Hackney resident remove his jacket in a bid to avoid detection and he was arrested days later on January 17.
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Tottenham released a statement at the time, which read: "We are appalled by the behaviour of a supporter that attempted to attack Arsenal goalkeeper Aaron Ramsdale at the end of today's match. Violence in any form has no place in football.
"The Club has reviewed its CCTV footage to identify the supporter and will be working with the Met Police, Arsenal and Aaron Ramsdale to take the strongest possible action, including an immediate ban from Tottenham Hotspur Stadium."
Speaking to Sky Sports after the match, Ramsdale said: "Spurs fans were giving me some crack second half, and I gave them some back.
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"The few people I did give it to it was probably well greeted, in a sportsmanship way.
"But then a fan jumped over and gave me a little punch in the back.
"That's what happened, and it's a shame because it is just a game of football at the end of the day.
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"Everyone, both sets of players, tried to bring me away. Thankfully nothing actually happened too drastically, but it's a sour taste."