Morata, who plays on loan at Juventus from Atletico Madrid, started on the bench at Wembley but came on to score an 80th minute equaliser against the Azzurri to cancel out Federico Chiesa's opener and take the semifinal to extra-time.
Alice Campello, Morata's wife, who is Italian, posted on Instagram stories some of the menacing messages she received written in Italian after the game.
"Your husband has cancer.... your children too" was one of the texts she received, while another one said: "Don't you dare post a photograph of Morata's goal or I will go to your house and burn you on live Instagram."
Campello responded to the abuse and wrote: "I'm honestly not suffering with regards to these messages, truthfully. I don't think it's a factor of 'Italians' but of ignorance. I do, however, think that if it had happened to a more fragile woman, it would have been a problem.
"Let's remember that this is a sport to unite not to vent your frustrations. I really hope that in the future there can be serious measures taken for these types of people because this is disgraceful and unacceptable."
Spain coach Luis Enrique publicly backed his player and said the threats should be "put in the hands of the police."
Morata, 28, received criticism throughout the tournament in Spain for his lack of effectiveness, yet his goal against Italy made him his country's top scorer in Euro finals with six goals, one more than Fernando Torres.
Following Spain's exit on Tuesday, Luis Enrique defended Morata and praised the player for coping with so much pressure.
"He's had to put up with many difficult things in this European Championship and has shown an enormous level," Luis Enrique said. "Morata had a physical problem, yet he took the penalty. I have to congratulate him for his personality."
In March, Morata addressed the importance of mental health for football players and revealed he came close to suffering depression during his first season at Chelsea and has since seen a psychologist to help him cope with the day-to-day pressure.