A club statement on Wednesday said: "The club adopts a responsible approach to social media and would expect any player connected to Hitchin Town to do likewise.
"In the light of the player's irresponsible and anti-social behaviour, we are terminating Alfie Barker's registration and relationship with the club with immediate effect."
The Football Association could also take a view on the issue given the governing body's guidelines on social network use.
Barker has since apologised for the tweets.
"I was watching Arsenal at home last night and I was hyper. At 3-0 down they weren't playing well and I was annoyed," he told The Comet in Hitchin.
"I'd had a couple of beers and was frustrated at the way the team was playing and I just lashed out. I have no idea why I focused on Harry Arter and his family.
"I am so, so sorry for what I have done. It was a moment of madness. I have no social skills. I have no filter between my mouth and my brain."
According to The Comet, Barker said he has lived with ADHD and a mild form of autism since being diagnosed five years ago.
"I work every day and I have spent the whole day trying to understand why I did what I did. I would like to meet Harry Arter face to face to apologise," he said,
"I cannot justify in any way, shape or form what I did but I would like to say sorry. I would like to write a letter to Harry saying sorry too.
"I feel like the worst person in the world today. It's the worst thing I've ever done in my life, but it is so out of character. I honestly don't know why I did it."
Arter has a tattoo on the inside of his right forearm which is dedicated to his daughter Renee who was stillborn on December 10 2015.
Speaking in a recent interview with the Guardian newspaper reflecting on his loss, Arter said: "Every day I think of her and it's strange because there are days when I feel like a normal person, or how someone who hasn't lost a baby would feel.
"But then it will just hit you, driving along and suddenly you see something that triggers your mind. It's such a weird feeling and one that I don't mind any more. Even when I'm sad I enjoy that feeling."
The 27-year-old's fiancee Rachel is currently pregnant and due to give birth to a baby girl in February.
Arter added: "I just hope everyone doesn't think that because we're having a new baby that we'll forget about Renee, because that's totally not the case. Rachel is still so sad about the situation, so I need to make sure that she stays strong now - which she is doing."