The Netherlands drew 1-1 after 90 minutes before going out 3-2 on penalties in Monterrey, with Justin Kluivert, Quinten Timber and Crysencio Summerville all missing their spot kicks before becoming targets of abuse online.
The Royal Dutch Football Association (KNVB) condemned the abuse in the strongest terms and confirmed it would seek criminal charges.
"We find this appalling," the KNVB said. "Once a report is filed, legal staff assess whether the statement constitutes a punishable offence.
"This can lead to a formal complaint being lodged with the Public Prosecution Service, which may then initiate a criminal investigation."
Koeman, 63, stepped down from his second spell as Netherlands head coach and hinted he may have ended his managerial career altogether.
He announced the decision on Instagram, where he also reflected on his wife Bartina's ongoing battle with breast cancer.
"Looking back on my career, I feel especially proud. I have worked with clubs and people that shaped me and gave me memories that I will cherish for a lifetime. We all dreamed of a World Cup in which we would write history. That didn't work out. No-one is more disappointed about that than me.
"The past years have made me realise again that there are more important things than football. Football has been my life, but health is priceless. When someone you love is fighting a tough battle, your perspective changes."
Koeman had one of the most distinguished careers in Dutch football history. As a player he won the European Cup with PSV in 1988 and Barcelona in 1992, scoring the winning goal in the Wembley final against Sampdoria from a free-kick. He earned 78 caps for the Netherlands and was part of the squad that won Euro 88.
His managerial career took in Vitesse, Ajax, Benfica, Valencia, PSV, Southampton, Everton and Barcelona before returning to the national team in 2023.
The Netherlands had reached the semi-finals of Euro 2024 before losing to England, but this World Cup campaign ended at the last-32 stage — a disappointing exit for a squad widely considered strong enough to go further.