Van Dijk is likely to miss the rest of the 2020/21 season Pickford's tackle damaged knee ligaments in October.
Pickford received no card for the reckless challenge after the Video Assistant Referee (VAR) had spotted an offside before the incident.
"It was a big game. Both teams were flying. The assistant has given offside and there is the delay in the flag, as we have had this season. Jordan then comes out to Virgil," Oliver told the Daily Mail.
"The thought initially was, 'It can't be a penalty because it's offside so we need to check the offside first'.
"I think I said to the VAR, 'If it's not offside, I'm going to give a penalty'.
"I have watched it back so many times. I genuinely don't think Pickford has done anything apart from try to spread himself but he did it the wrong way, as the injury has shown.
"We have all, myself included, not thought about the challenge as much as we should have done. We could still have given offside and sent Pickford off.
"What I was surprised about looking at it afterwards was that nothing was expected on-field in terms of a red card. None of the players were asking for that.
"We got sucked too much into going step by step as opposed to thinking of the bigger process, which was considering the challenge as well and not just the fact it can't be a penalty.
"We should have restarted wth the offside, as we did, but with a different punishment for Jordan Pickford."