Liverpool midfielder Georginio Wijnaldum has described Pickford’s challenge as stupid and said the goalkeeper “didn’t care what happened” when he went to tackle Van Dijk.
Liverpool said Van Dijk will undergo surgery on the knee injury he sustained during Saturday's 2-2 draw against Everton.
Liverpool did not give a timescale on Van Dijk's absence, but an injury of that nature will likely keep the defender out for a long period.
Liverpool extended its 10-year unbeaten record against its neighbor but Klopp was a frustrated man as he made the short trip home from Goodison.
Everton clung on to their unbeaten start to the season to remain three points clear at the top of the Premier League as a number of controversial calls went their way.
Just when the visitors looked like claiming a point, new signing Rodrigo - on as a substitute - fouled Fabinho in the area and Salah sidefooted home the resulting spot-kick.
'Shootouts like 'lottery': Virgil van Dijk urges Liverpool not to 'panic' after Community Shield defeat
Arsenal had defeated Liverpool 5-4 on penalties to win the Community Shield on Saturday at the Wembley Stadium.
During a friendly match against RB Salzburg on Tuesday, the centre-back was forced off in the 55th minute at Red Bull Arena, where Rhian Brewster struck twice in the second half for the Reds.
Liverpool manager Jurgen Klopp hailed Virgil van Dijk's mindset and said he had no idea that the defender would be 'so special'.
Liverpool lead the Premier League table by 25 points with nine games remaining as they chase their first title in 30 years.
Van Dijk was pitted against Messi during last season's Champions League semi-final, as Liverpool famously overcame a 3-0 first-leg deficit by triumphing 4-0 at Anfield to book a spot in the finals.
Liverpool top the Premier League table with 82 points, 25 points ahead of the second-placed club Manchester City.
Despite the defeat, Liverpool top the Premier League table with 79 points, 22 points ahead of the second-placed Manchester City.
Van Dijk has been so instrumental in Liverpool's improvement from the side that finished 25 points adrift of City last season.
Mourinho has mischievously highlighted the changing view of Jurgen Klopp after his Liverpool counterpart spent £75 million to sign the defender.