Van Dijk sustained a cruciate ligament injury after a strong challenge from Toffees shot-stopper Pickford in the first half of the Merseyside derby on Saturday.
The Reds star will potentially have to remain on the sidelines for the rest of the current season and is scheduled to undergo surgery on his knee.
Ancelotti sent his own, Pickford’s and his club’s best regards to the Dutchman, wishing him a speedy recovery.
The Italian admitted that Pickford’s tackle was mistimed, but stressed the Englishman did not hold any ill will against Van Dijk as the unfortunate incident was a result of two players putting their bodies on the line for their teams.
“To be clear, we are really sorry for the injury of Virgil van Dijk”, Ancelotti told Everton TV.
“Every one of us hopes he can recover soon and well.
“There was this contact with Jordan Pickford: it was mistimed contact.
“But the fact the Premier League is so fast, it is not so difficult to arrive a little bit later.
“Jordan arrived a little bit later but his intention was to try to reach the ball.
“It was not to hurt Virgil van Dijk.
“To say it was premeditated, and this and that, in my opinion, is too much.
“Virgil van Dijk knows this.
“Jordan is really sad, really disappointed [that Van Dijk is injured].
“It was a strong contact, maybe mistimed, but sometimes in football it can happen.”
Liverpool managed to keep a clean sheet in their first full game without Van Dijk after registering a 1-0 win away at Dutch giants Ajax in the Champions League on Wednesday.