Nigel Dewale sent two posts from his TikTok account under the username Bogeyman during the tournament, held in Switzerland.
He used derogatory language, saying people with brown skin were murderers and "groomers".
The messages were later traced to Dewale, who was arrested and charged.
He was sentenced at Blackburn Magistrates' Court, in northwest England, to six weeks in prison, suspended for 12 months.
Dewale pleaded guilty at an earlier hearing to sending a grossly offensive message via a public communications network.
Former Chelsea defender Carter, 28, who plays for the US team Gotham FC, was part of the Lionesses squad that successfully defended their title at the European Championship.