Gareth Southgate’s side were awarded a controversial penalty when Harry Kane came into contact with Denzel Dumfries.
The on-field officials originally waved away any protests, but VAR intervened to ask Felix Zwayer to go and take a look at the monitor.
He subsequently overturned his decision, meaning Kane could score a crucial goal that turned the game on its head.
“I think it says it all that the referee went in quite quickly after the game. I had no time to shake his hand.
“But listen; it is what it is. The game is done. We lost. It’s very tough to take.
“Certain moments, it was obvious that it should have gone our way. It didn’t. It’s always tough.
“I don’t know what to say. It’s just difficult to accept this. I’m full of emotions. It’s been a very, very long year, a tough year.
“We had a big dream and we felt like we could have achieved that here as a team. Unfortunately it didn’t happen.”
The captain was also pressed on whether officials should be made to explain their decision-making.
“I don’t know,” he continued. “They keep changing certain things. Small changes that could have big impacts, but maybe it’s a good thing that they could be held accountable as well.
“Let them come here and speak to you guys and explain themselves in certain moments, like we have to explain ourselves when we do something wrong as well. I think that might be something.
“But I shouldn’t speak about others. We have to look at ourselves and unfortunately we are out.”
The Dutch were originally on course for a date with Spain in Munich on Sunday after taking a 1-0 lead through Xavi Simons’ early strike.
But their hearts were broken not only by Kane’s goal, but also Ollie Watkins’ last-gasp winner.
Van Dijk added: “I think, throughout the whole tournament, we have seen so many sides of our team.
“Good sides in the attacking form, the defensive form some games. And also the bad side for us against Austria.
“But today was a good game. It was a good game for many reasons, with and without the ball.
“It was an intense game. They created good moments. We had good moments.
“But to concede a goal in the last minute it always difficult to accept, especially when it is a knock-out and a semi-final and it shatters our dreams in the last second of the game.”