"Whilst every fan is entitled to their opinion on performance and result I don't agree or think that it's OK to target someone's appearance or race," Carter posted in a lengthy Instagram statement.
"I am taking this measure to protect myself in a bid to keep my focus on helping the team anyway I can.
"Hopefully speaking out will make the people writing this abuse think twice so that others won't have to deal with it."
The 27-year-old has started all four of England's games at the tournament, and they next play Italy in the semi-finals on Tuesday in Geneva.
"We have made historic changes with this Lionesses squad that I am so proud to be a part of and my hope is that by speaking out about this it will make another positive change for all."
FA CEO Mark Bullingham condemned the abuse.
"Our priority is Jess and giving her all the support she needs," Bullingham said in a statement. "We strongly condemn those responsible for this disgusting racism."
Bullingham said the FA has been in contact with UK police, who are in touch with the relevant social media platform.
"We are working with police to ensure those responsible for this hate crime are brought to justice," he said.
"Regrettably this is not the first time this has happened to an England player, so we had measures in place to allow us to respond quickly and where possible provide information to support any potential police action."
While the England women traditionally take a knee before matches - including their four matches so far in Switzerland, as a symbolic gesture against racism - they will not do so on Tuesday.
"It is clear we and football need to find another way to tackle racism, we have agreed as a squad to remain standing before kickoff on Tuesday," they said in a statement.
"We stand with Jess and all Lionesses players past and present who have suffered racism," the team said. "Representing your country is the greatest honour. It is not right that while we are doing that some of us are treated differently simply because of the colour of our skin."
Europe's football governing body UEFA said: "Abuse and discrimination should never be tolerated, whether in football or society, in person or online.
"We stand with Jess."