The suspect used to work at the home, near Montpellier, a source close to the investigation told AFP news agency.
Prosecutor Christophe Barret said the motive was not clear but it was not linked to "Islamist terrorism".
France remains in a formal state of emergency since a wave of terrorist attacks last year.
The suspect, 47, was arrested near his home, about 10 miles (15km) from Montferrier-sur-Lez, the town where the attack was carried out.
He worked at the home several years ago, and is now unemployed, the source said.
Witnesses said the assailant was armed with a shotgun and a knife when he entered the home on Thursday night.
A nurse raised the alarm after escaping from the home but when police arrived they found the body of the dead woman gagged and tied up outside the building.
The 54-year-old woman, who worked in the home, has not been named.
The home caters for former missionaries with past service in Africa, as well as several nuns.
Residents "are very elderly with an average age of 75 although some are over 90", a local councillor said, adding that many need assistance to walk.
A former soldier, suspected of stabbing a woman to death in a retirement home for former missionaries, has been arrested in France.