The woman had been asked by her daughter to take down the pictures from Facebook and Pinterest several times, but she did not respond -- so the family dispute ended up in court.
Publishing the children's pictures on social media would, according to the mother, "seriously violate their privacy," the court ruling states.
The Gelderland judge agreed that the grandmother did not have permission to post the pictures under General Data Protection Regulation (GDPR) legislation.
Those rules do not normally apply to the storage of personal data within personal circles such as family.
However, in this case, the grandmother had made the photos public without the consent of the mother -- who has legal authority over which data of her underage children may be stored and shared.
Furthermore, by posting of photographs on social media, the grandmother made them available to a wider audience, the court's ruling, published earlier this month, explained.
"On Facebook, it cannot be ruled out that placed photos could be distributed and that they may come into the hands of third parties," the ruling said.
The grandmother must remove the pictures of her grandchildren from Facebook and Pinterest within ten days, the judge ruled. If she does not, she must pay a penalty of €50 ($55) per day that the photos are online, with a maximum penalty of €1,000 ($1,100).
The daughter had asked to impose a penalty of €250 ($275) per day if the photos remained.
GDPR is the European Union's data privacy law, which came into effect in 2018.
It gives people more control over their personal data and forces companies to make sure the way they collect, process and store data is safe.
The EU's intention was to achieve a fundamental change in the way companies use data -- with its central idea being that people are entitled "privacy by default."