The 3-2 midweek win for the Toffees secured their Premier League status and sparked a pitch invasion at Goodison Park, the latest in a long line to occur in England after similar scenarios at Nottingham Forest, Northampton and Port Vale during the past week.
As Vieira was walking off the pitch to the away dressing room, he appeared to be repeatedly goaded by a fan and then reacted by attempting to grab and kick out at the supporter.
Further footage has since emerged of Wilfried Zaha and Ebere Eze being confronted by fans on the pitch and the Football Association is looking into it while Merseyside Police confirmed an investigation has been opened.
Reflecting after a final-day win over Manchester United, Vieira said: “I think the Everton incident is bigger than just that incident.
“When you look at it in the Premier League, there were a lot of field invasions that can cause trouble. There are real issues here to reflect on and try to see what is the best way to manage it and to deal with it.
“You don’t know what can happen. When you don’t know what can happen of course you fear for anything, for the players.
“We want to be safe and the best way to do that is we need to reflect on what is the best way for those kind of things not to happen on the field. As a coach, manager, player or staff we want to be safe in our workplace.
“I think there is a big issue on the FA and Premier League’s plate to deal with.
“They need the support and the club and the players to get involved. Everybody has to take responsibility and see what is the best way for those things not to happen.”
Vieira revealed he would be happy to speak to the FA about his own incident but stressed England faced a country-wide problem.
Sheffield United’s Billy Sharp was headbutted last Tuesday by Forest season-ticket holder Robert Biggs, who has since been sentenced to 24 weeks in jail whilst players from Mansfield and Swindon were involved in incidents at play-off ties at Northampton and Port Vale.
Meanwhile, Aston Villa manager Steven Gerrard revealed after their 3-2 loss away to Man City, which earned the hosts the title, that his goalkeeper Robin Olsen had been attacked following a pitch invasion at the Etihad.
“I will participate with the FA and talk about the incident and give my opinion and my view,” Vieira added.
“Again, it is not about just that incident. It is about all the incidents and I think it happened too often in the last couple of weeks and we need to reflect on that and make a decision so players, staff and fans can be safe going to watch football matches.”