Firefighters were called to the woman's apartment in a public housing complex at 1152 Turk St. in the city's Western Addition neighborhood at about 1:15 p.m., spokeswoman Lt. Mindy Talmadge said.
The fire was extinguished by the apartment's sprinkler system and confined to one room, Talmadge said.
By the time firefighters and police arrived, the man had left.
He was later arrested at another location, and then hospitalized for an undisclosed medical condition, police said.
Police spokesman Officer Albie Esparza said the man and the woman were friends and that she had let him stay in her apartment for the past several months.
On Tuesday, the pair got into an argument and the 43-year-old woman told her friend that "he was no longer welcome to stay there," Esparza said.
The woman left the apartment at about 12:40 p.m. and about half an hour later got a call from a neighbor that her home was on fire, Esparza said.
Damage to the home was estimated at $40,000, Talmadge said.