Gini Wijnaldum says Everton take their aggression "way too far" in Merseyside derbies, following the game at the weekend which saw two shocking tackles on his team-mates.
The damage done to Van Dijk's knee was severe, as he will now miss the majority of the season.
Commenting on that, Wijnaldum said Pickford was "stupid".
"Of course we are upset. The way Pickford went into the challenge was completely, in my opinion, stupid," he told Sky Sports.
"I believe that he didn't want to injure Virgil the way he injured him but the way he took the tackle, he didn't care about what happened after the tackle."
In the latter stages of the derby, which finished 2-2, Richarlison was sent off for another late tackle, with Thiago being the victim on this occasion.
Wijnaldum believes that this type of physicality is typical of Everton in the game between the city rivals, and that they sometimes take it too far.
"I think we had it a lot with the games against Everton. In my opinion, they take it way too far in the games we play against them," he added.
"We know it's a derby and everyone wants to win the derby but they took it a little bit over the top.
"The tackle from Richarlison on Thiago was also a nasty tackle. The way they were doing it [tackling] was completely unacceptable. It makes it harder that they will not be punished."
With Joel Matip also out for that game, and Van Dijk unavailable for the foreseeable future, Jurgen Klopp has some selection dilemmas in defence.