Farnborough Football Club also fined the defender two weeks' wages after he admitted writing the comments about race, disability and paedophiles.
In a statement, the Conference South club said that his behaviour was "unacceptable in any walk of life".
The club added that it had taken action against Bygrave, who has previously played for Reading, GIllingham, Weymouth and Hayes & Yeading, after it was alerted to accusations against him last week.
"Adam has accepted... that he has made numerous racist, disabled and pedophile based comments on social networking sites which are totally unacceptable in any walk of life," the statement said.
"The club have acted swiftly having only found out of the accusations late last week, taken legal advice, spoken with the Football Association and confronted the player today.
"Farnborough Football Club is committed to its community roots and it will not accept such behaviour or actions from any of its employees or supporters, we do believe though that this is a totally isolated incident."
Bygrave said in his own statement that he "deeply regretted" making the comments, which he felt were just "banter".
"As much as I totally accept that they are inappropriate and will offend many people, I believed I was just repeating banter and jokes from numerous sources," he said.
"I accept the severe punishment handed out by the football club as I have left them with no option.
"I hope people will accept that I have made a mistake, and just because I have posted comments of this nature, it does not mean that I believe in them."
Bygrave's sacking comes at a time when footballers are under the spotlight for their use of social media services such as Facebook and Twitter.
Earlier in the month, Manchester United defender Rio Ferdinand was fined £45,000 after being found guilty of improper conduct over a "choc ice" tweet about fellow footballer Ashley Cole.