The game, which was scoreless at the time, was halted in the 12th minute while medical staff dealt with the emergency, and the referee told players to return to the dressing rooms six minutes later.
An announcement was made almost an hour later that the match would not resume, and Charlton later said the fan had died after being taken to the hospital.
"The club are devastated to report that a Charlton Athletic supporter has died following a medical emergency during today’s game at The Valley against Portsmouth," their statement said.
"The club are grateful to supporters in attendance and the Portsmouth players and staff for their patience and understanding as this tragedy unfolded.
"Everyone at the club sends their heartfelt condolences to the supporter's family and friends at this incredibly difficult time."