The Netherlands international picked up a knock in Roma's 1-0 defeat against Napoli on Sunday and was forced to leave the pitch after just 12 minutes of play.
Strootman underwent an MRI scan on Monday morning to determine the extent of the knee problem and Van Gaal has confirmed the worst.
He told www.knvb.nl: "This is dramatic for Kevin. I can imagine that his world collapsed.
"Kevin stood with both his club AS Roma, as with Orange 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."
Roma later confirmed Strootman, 24, had damaged his anterior cruciate ligament and possibly his meniscus.
A club statement read: "Kevin Strootman had an MRI scan this morning as he was examined by Professor Giuliano Cerulli.
"Tests showed he has torn the anterior cruciate ligament in his left knee and there could also be damage to the meniscus. He will undergo surgery in the next few days."
He has developed into a key player at the capital club since joining them from PSV in the summer of 2013, 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."