Netherlands striker Van Persie, 33, had scored the opening goal in the Turkish side's 3-1 win against Akhisar before the first-half injury.
Television pictures showed blood coming through a bandage on his left eye.
"After tests in the hospital this morning I can say my eye itself is undamaged so all good," he said.
He told followers on Instagram that he had torn his eyelid.
Club doctor Burak Kunduracioglu had earlier sought to play down fears the player had suffered a serious eye injury, saying: "He received a blow to the eye and suffered a torn eyelid and bleeding.
"After the investigations we learned that nothing important happened. His condition is good."
The former Arsenal and Manchester United forward joined Fenerbahce from the Old Trafford club in July 2015.
Fenerbahce's Robin van Persie has reassured supporters that his eye was not damaged after he was carried off the pitch bleeding on Sunday.