It ended up in court after a falling-out between the woman and her daughter.
The judge ruled the matter was within the scope of the EU's General Data Protection Regulation (GDPR).
One expert said the ruling reflected the "position that the European Court has taken over many years".
The case went to court after the woman refused to delete photographs of her grandchildren which she had posted on social media.
The mother of the children had asked several times for the pictures to be deleted.
The GDPR does not apply to the "purely personal" or "household" processing of data.
However, that exemption did not apply because posting photographs on social media made them available to a wider audience, the ruling said.
"With Facebook, it cannot be ruled out that placed photos may be distributed and may end up in the hands of third parties," it said.
The woman must remove the photos or pay a fine of €50 (£45) for every day that she fails to comply with the order, up to a maximum fine of €1,000.
If she posts more images of the children in the future, she will be fined an extra €50 a day.
"I think the ruling will surprise a lot of people who probably don't think too much before they tweet or post photos," said Neil Brown, a technology lawyer at Decoded Legal.
"Irrespective of the legal position, would it be reasonable for the people who've posted those photos to think, 'Well, he or she doesn't want them out there anymore'?"
"Actually, the reasonable thing - the human thing to do - is to go and take them down."