The Metropolitan Police named the man as 44-year-old Michael Dye from Cardiff and said that six people, understood to be Welsh, had been arrested outside the ground before Tuesday's Group G match which England won 1-0.
"Police were called at approximately 19:20 (18:20 GMT)... to an area outside Wembley Stadium following reports of an assault," a police statement said.
"Michael John Dye... suffered head injuries and was taken by London Ambulance Service to a central London hospital. He was pronounced dead at 20:50."
Local media quoted paramedics as saying Dye had also suffered a heart attack before being taken to hospital.
Wales coach Gary Speed offered his condolences to Dye's family and agreed the incident had overshadowed the match.
"It's a game of football at the end of the day and there's nothing more precious than life," he told Sky Sports News. "The players and the staff, we're all deeply saddened by the news that we found out after the game."
Dye was a Wales and Cardiff City fan, according to a statement on the Championship club's website, while local media reported he had worked as a mason for his city's council.
"Out of respect for the family of the supporter, the sensitivity of the situation and until full details and circumstances are ascertained, no statements will be made until deemed fit by proper channels," the Cardiff club said.
