Oyston has been accused of making abusive remarks about disabled people to Tangerine Knights Supporters Group member Steve Smith.
Smith was critical of the manner in which Oyston is running the Championship club and the chairman is accused of responding by calling him a "massive retard" and an "intellectual cripple."
Oyston issued a statement on Monday apologising for his actions and pledged to make a donation to the club's Community Trust.
The statement said: "I would like to unreservedly apologise for any offence or distress caused by my text responses reported in the media recently. I regret stooping to the level of those threatening and abusing my family.
"My mobile number was placed on a social media website recently and it led to a barrage of abuse about my mother, father, wife, children and myself. In hindsight the aim of this was clear, and I foolishly opted to challenge some of the abuse, harassment and threats.
"Given my position, I should have acted with more responsibility and reported the abusive text messages, phone calls and answer phone messages to the police, rather than get embroiled in such an exchange.
"At Blackpool FC we pride ourselves on our work in the community, and I wouldn't want my unacceptable comments to detract from the fantastic work being done by our Community Trust.
"The Community Trust works with a number of organisations to support making as many opportunities for people with disabilities, including Warren Manor Day Care Centre, Age UK, the Wheelchair Football Association and Sport England, as well as special schools on the Fylde Coast."
Blackpool are bottom of the Championship and were hammered 6-1 by leaders Bournemouth at Bloomfield Road on Saturday.
