The Netherlands midfielder suffered a ruptured ACL and torn meniscus in Roma's 1-0 defeat to Napoli on Sunday.
Strootman suffered the injury in Roma's 1-0 defeat to Napoli on Sunday and was forced to leave the pitch after just 12 minutes.
Strootman underwent an MRI scan Monday morning to determine the extent of the knee problem and Van Gaal has confirmed the worst.
He told KNVB.nl: "This is dramatic for Kevin. I can imagine that his world collapsed. Kevin stood with both his club AS Roma, as with Oranje for a very important period. Kevin had a very big contribution to our qualification for the World Cup in Brazil. But the worst is for Kevin, who sees his dreams are shattered."
It has been reported Strootman, 24, has a ruptured anterior cruciate ligament as well as a torn meniscus in his left knee. Not only will he miss the World Cup but also the remainder of the Serie A season, which represents a huge blow for Roma.
He has developed into a key player at the capital club since joining it from PSV in the summer, scoring five goals in 24 league appearances.
Strootman was also one of only three players guaranteed a spot in Oranje's World Cup squad after Van Gaal recently branded him "untouchable" alongside Robin van Persie and Arjen Robben.
Netherlands midfielder Nigel de Jong was quick to offer his support, tweeting: "Wish @Kevin_strootman all the best with his recovery. Keep your chin up and return stronger than before! Non mollare Mai! #dedication."