The Dutch international and 2018/19 UEFA Men's Player of the Year was forced off in the early stages of the match after he was wildly clattered into by Toffees goalkeeper Jordan Pickford, who escaped punishment.
New signing Thiago Alcantara was also on the receiving end of a similarly clumsy challenge from Richarlison, with referee Michael Oliver showing the Brazilian a straight red card for his involvement.
Liverpool are currently awaiting the results of a scan to confirm the severity of van Dijk's injury, and look set to be forced into preparing for life without their star defender for at least a few weeks.
Speaking about the incident in the 11th minute of yesterday's Merseyside derby, Robertson said: "To be honest, I never saw it. I just saw Virg going down.
"But the lads when they came in at half-time said it was a shocking challenge, so that’s two they’ve done us with.
"I can’t comment on if Pickford’s meant it or not, I think he’s an honest lad and I don’t think he would have.
"But it doesn’t look good for Virg and I hope he comes back as quickly as he can, and hopefully the scan maybe doesn’t show what the lads think."
It remains to be seen who Jurgen Klopp will consider to replace the 29-year-old, who has established himself as one of the world's finest central defenders.
“Losing a player like him is always tough, but from an incident like that, it makes it a wee bit tougher," Robertson continued.
“Look, that’s part of football unfortunately. It’s part of derbies, it’s part of life.
"We’ll all deal with it, we’ll all support Virg for however long he’s out – hopefully not long, like I said."
Saturday's game was also marred by a hugely controversial VAR decision that ruled out a Jordan Henderson winner deep into stoppage time.
Sadio Mane was ruled to have been marginally offside in the build-up, much to the frustration of everybody associated with the defending Premier League champions.
Reflecting on the result, Robertson said: "It’s obviously disappointing and I think we’ll be the more disappointed team.
“The game was there for us, we created enough chances. We thought we had nicked it at the end and then I think Sadio’s toenail was maybe offside but that’s what VAR brings.
"Onside, offside, it’s clear, so unfortunately it doesn’t matter the margin, he’s offside."
“It’s always tough to take when you think you’ve won it and you are that close and things like that. But we take a point and we move on."