The 31-year-old, who was part of the Storm side that reached the grand final last year, played for 25 minutes in a 28-24 National Rugby League loss at North Queensland on Saturday.
Kamikamica experienced a suspected stroke early on Monday morning, said the Storm, and was taken to hospital by ambulance.
"Melbourne Storm can confirm that forward Tui Kamikamica has been discharged from hospital to commence a period of inpatient rehabilitation," said a statement on Thursday.
"It is anticipated he will be able to return to the club following his rehabilitation period and continue his recovery under the care of Melbourne Storm medical staff.
"The club would like to thank the medical professionals at Royal Melbourne Hospital and Ambulance Victoria for their outstanding care of Tui and extend its appreciation to everyone who has supported Tui and his family during this difficult time."
Kamikamica has played 142 NRL games, all for the Storm, since his debut in 2017 and has made 22 appearances for Fiji.
He was due to captain Fiji in this year's rugby league World Cup, which begins in October in Australia, New Zealand and Papua New Guinea.
Melbourne are already without fellow forward Eli Katoa for the season after he underwent brain surgery following a series of head knocks playing for Tonga against New Zealand in November.