The statement said the couple were "a few months into a relationship" and it was "not right" that Ms Markle should be subjected to such treatment.
It said the prince rarely took formal action over "fictional stories".
"But the past week has seen a line crossed," it added.
In recent days a number of newspapers have carried front page stories about the actress, best known for playing Rachel Zane in the TV drama Suits.
'Substantial bribes'