Oyston admitted sending the "unacceptable" texts in response to "a barrage of abuse" he received after his mobile number was placed on a social media website.
"I regret stooping to the level of those threatening and abusing my family," said Oyston in a statement.
"Given my position, I should have acted with more responsibility."
Oyston, a member of the Football League's board of directors, is alleged to have called Stephen Smith "a retard" and told the individual to "enjoy the rest of your special needs day out".
During the text exchange, Oyston used numerous swear words and called Smith "educationally subnormal" and a "sad act".
He is also reported to have told Smith that he has banned him from the club's Bloomfield Road ground.
The Football Association has said it is aware of the row.
In his statement, Oyston said: "I would like to unreservedly apologise for any offence or distress caused by my text responses reported in the media recently.
"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."
Oyston said that he should have "reported the abusive text messages, phone calls and answer phone messages to the police, rather than get embroiled in such an exchange".
He also said he would make a donation to the club's Community Trust, which works with a number of organisations that support disabled people.
Commenting on Oyston's statement, Smith told BBC Radio 5 live: "I think the apology from Karl is weak."
Smith claimed his texts to Oyston were "originally about the club or lack of financial investment in the club infrastructure", "the weak squad" and the "position that Blackpool football club finds itself in".
He added: "Regardless of whether Karl's annoyed at his number being released or not, he's still a Football League chairman and as such he should behave accordingly.
"I know there are messages between him and other fans. I can't comment personally on whether one of those fans threatened him or his family, but I certainly wasn't one of those."
The Blackpool Supporters Trust (BST) criticised Oyston after the texts were made public, while the Blackpool Gazette newspaper decided to drop him as a weekly columnist.
"We would support the Oystons if they could run the club in a responsible manner, but recent evidence suggests they are not able to do so," said Tim Fielding, chair of the BST.
"I am astonished, really. It's quite remarkable. For the owner of a football club to say that is bad judgement.
"He is a flag-bearer for the town, for the club and the community. It's a sad day for Blackpool Football Club."
The club are bottom of the Championship and lost 6-1 at home to leaders Bournemouth on Saturday.