In a message in Persian posted with a picture of Karrubi, Eyre writes that it was with "with deepest regret" that he found out about the death of Fatemeh Karrubi, the sister of the former Iranian parliament speaker.
The Facebook message is a small gesture by the State Department toward Iran's opposition movement, which has been under intense pressure from the country's leaders.
Iranian authorities have been holding Karrubi under house arrest with reportedly almost no contact with the outside world in an apparent effort to silence him and make the public and his supporters forget him.
In a letter issued last week, Karrubi's wife wrote that the 74-year-old cleric had recently undergone back surgery. She said Karrubi had been hospitalized in Tehran's Arad hospital for nine days.
Karrubi's family has expressed concern over his health, which they say has deteriorated under house arrest. Karrubi is reportedly being held at an intelligence ministry safe house.
Opposition leader Mir Hossein Musavi along with his wife university professor Zahra Rahnavard are also being held under house arrest.
The three opposition figures were put under house arrest in February 2011 after calling for a demonstration in solidarity with uprisings in Arab countries. Tens of thousands of opposition supporters responded.
Since the election of Iran's President Hassan Rohani, calls for their release have increased.
In his election campaign, Rohani had promised to release political prisoners.
The social-media-savvy Eyre had previously used Facebook to send his condolences over the death of the mother of Iran's Foreign Minister Javad Zarif.
At the time, some users criticized Eyre for reaching out to an Iranian official.